Is it possible to form a triangle with the given
side lengths? If not, explain why not.
1.5 cm, 7 cm, 10 cm
SOLUTION:
The sum of the lengths of any two sides of a triangle
must be greater than the length of the third side.
Yes; 5 + 7 > 10, 5 + 10 > 7, and 7 + 10 > 5
ANSWER:
Yes; 5 + 7 > 10, 5 + 10 > 7, and 7 + 10 > 5
2.3 in., 4 in., 8 in.
SOLUTION:
No; . The sum of the lengths of any two
sides of a triangle must be greater than the length of
the third side.
ANSWER:
No;
3.6 m, 14 m, 10 m
SOLUTION:
Yes;6+14>10,6+10>14,and10+14>6.
The sum of the lengths of any two sides of a triangle
must be greater than the length of the third side.
ANSWER:
Yes; 6 + 14 > 10, 6 + 10 > 14, and 10 + 14 > 6
4.MULTIPLE CHOICE If the measures of two
sides of a triangle are 5 yards and 9 yards, what is
the least possible measure of the third side if the
measure is an integer?
A 4 yd
B 5 yd
C 6 yd
D 14 yd
SOLUTION:
Let x represents the length of the third side. Next, set
up and solve each of the three triangle inequalities.
5 + 9 > x, 5 + x > 9, and 9 + x > 5
That is, 14 > x, x > 4, and x > –4.
Notice that x > –4 is always true for any whole
number measure for x. Combining the two remaining
inequalities, the range of values that fit both
inequalities is x > 4 and x < 14, which can be written
as 4 < x < 14. So, the least possible measure of the
thirdsidecouldbe5yd.
The correct option is B.
ANSWER:
B
PROOF Write a two-column proof.
5.Given:
Prove:
SOLUTION:
Think backwards when considering this proof. Notice
that what you are trying to prove is an inequality
statement. However, it isn't exactly related to
, except for instead of side being used, it
is Since it is given that , you can
easilyusethisinasubstitutionstep.
Given:
Prove: YZ + ZW > XW
Statements (Reasons)
1. (Given)
2. XW = YW (Def. of segments)
3. YZ + ZW > YW ( Inequal. Thm.)
4. YZ + ZW > XW (Substitution Property.)
ANSWER:
Given:
Prove: YZ + ZW > XW
Statements (Reasons)
1. (Given)
2. XW = YW (Def. of segments)
3. YZ + ZW > YW ( Inequal. Thm.)
4. YZ + ZW > XW (Subst.)
Is it possible to form a triangle with the given
side lengths? If not, explain why not.
6.4 ft, 9 ft, 15 ft
SOLUTION:
No; . The sum of the lengths of any two
sides of a triangle must be greater than the length of
the third side.
ANSWER:
No;
7.11 mm, 21 mm, 16 mm
SOLUTION:
Yes;11+21>16,11+16>21,and16+21>11.
The sum of the lengths of any two sides of a triangle
must be greater than the length of the third side.
ANSWER:
Yes; 11 + 21 > 16, 11 + 16 > 21, and 16 + 21 > 11
8.9.9 cm, 1.1 cm, 8.2 cm
SOLUTION:
No; The sum of the lengths of any two
sides of a triangle must be greater than the length of
the third side.
ANSWER:
No;
9.2.1 in., 4.2 in., 7.9 in.
SOLUTION:
No; The sum of the lengths of any
two sides of a triangle must be greater than the
length of the third side.
ANSWER:
No;
10.
SOLUTION:
No; The sum of the lengths of any two
sides of a triangle must be greater than the length of
the third side.
ANSWER:
No;
11.
SOLUTION:
Yes;
The sum of the lengths of any two sides of a triangle
must be greater than the length of the third side.
ANSWER:
Yes;
Find the range for the measure of the third side
of a triangle given the measures of two sides.
12.4 ft, 8 ft
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than 4 +
8. Therefore, n < 12.
If n is not the largest side, then 8 is the largest and 8
must be less than 4 + n. Therefore, 4 < n.
Combining these two inequalities, we get 4 < n < 12.
ANSWER:
4 ft < n < 12 ft
13.5 m, 11 m
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than 5
+11. Therefore, n < 16.
If n is not the largest side, then 11 is the largest and
11 must be less than 5 + n. Therefore, 6 < n.
Combining these two inequalities, we get 6 < n < 16.
ANSWER:
6 m < n < 16 m
14.2.7 cm, 4.2 cm
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than 2.7 +
4.2. Therefore, n < 6.9.
If n is not the largest side, then 4.2 is the largest and
4.2 must be less than 2.7 + n. Therefore, 1.5 < n.
Combining these two inequalities, we get 1.5 < n <
6.9.
ANSWER:
1.5 cm < n < 6.9 cm
15.3.8 in., 9.2 in.
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than 3.8 +
9.2. Therefore, n < 13.
If n is not the largest side, then 9.2 is the largest and
9.2 must be less than 3.8 + n. Therefore, 5.4 < n.
Combining these two inequalities, we get 5.4 < n <
13.
ANSWER:
5.4 in. < n < 13 in.
16.
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than
. Therefore, .
If n is not the largest side, then is the largest and
must be less than . Therefore, .
Combining these two inequalities, we get
.
ANSWER:
17.
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than
. Therefore, n < .
If n is not the largest side, then is the largest and
must be less than . Therefore, .
Combining these two inequalities, we get
.
ANSWER:
PROOF Write a two-column proof.
18.Given:
Prove:
SOLUTION:
The key to this proof is to figure out some way to get
BC=BD so that you can substitute one in for the
other using the Triangle Inequality Theorem.
Consider the given statement, if two angles of a
triangle are congruent, what kind of triangle is it and ,
therefore, how do you know that BC must equal BD?
Proof:
Statements (Reasons)
1. (Given)
2. (Converse of Isosceles Thm.)
3. BC = BD (Def. of segments)
4. AB + AD > BD ( Inequality Thm.)
5. AB + AD > BC (Substitution Property.)
ANSWER:
Proof:
Statements (Reasons)
1. (Given)
2. (Conv.Isos. Thm.)
3. BC = BD (Def. of segments)
4. AB + AD > BD ( Inequal. Thm.)
5. AB + AD > BC (Subst.)
19.Given:
Prove: KJ +KL> LM
SOLUTION:
Think backwards when considering this proof. Notice
that what you are trying to prove is an inequality
statement. However, it isn't exactly related to
, except for instead of side being used, it
is Since it is given that , you can
easily use this in a substitution step
Proof:
Statements (Reasons)
1. (Given)
2. JL = LM (Def. of segments)
3. KJ + KL > JL ( Inequality Thm.)
4. KJ + KL > LM (Substitution Property)
ANSWER:
Proof:
Statements (Reasons)
1. (Given)
2. JL = LM (Def. of segments)
3. KJ + KL > JL ( Inequal. Thm.)
4. KJ + KL > LM (Subst.)
CCSS SENSE-MAKINGDetermine the
possible values of x.
20.
SOLUTION:
Set up and solve each of the three triangle
inequalities.
Notice that isalwaystrueforanywhole
number measure for x.The range of values that
would be true for the other two inequalities is
and , which can be written as
.
ANSWER:
6 < x < 17
21.
SOLUTION:
Set up and solve each of the three triangle
inequalities.
Notice that isalwaystrueforanywhole
number measure for x. Combining the two remaining
inequalities, the range of values that fit both
inequalities is and , which can be
written as .
ANSWER:
22.DRIVING Takoda wants to take the most efficient
route from his house to a soccer tournament at The
Sportsplex. He can take County Line Road or he can
take Highway 4 and then Route 6 to the get to The
Sportsplex.
a. Which of the two possible routes is the shortest?
Explain your reasoning.
b. Suppose Takoda always drives below the speed
limit. If the speed limit on County Line Road is 30
miles per hour and on both Highway 4 and Route 6 it
is 55 miles per hour, which route will be faster?
Explain.
SOLUTION:
a. County Line Road; sample answer: In a triangle,
the sum of two of the sides is always greater than
the third side, so the sum of the distance on Highway
4 and the distance on Route 6 is greater than the
distance on County Line Road. Or you can add the
distances using Highway 4 and Route 6 and compare
their sum to the 30 miles of County Line Road. Since
47 miles is greater than 30 miles, County Line Road
istheshortestdistance.
b. Highway 4 to Route 6; sample answer: Since
Takoda drives below the 30 mph speed limit on
County Line Road and the distance is 30 miles, it will
take him about 30/30 = 1 hour to get to The
Sportsplex. He has to drive 47 miles on Highway 4
and Route 6, and the speed limit is 55 miles per hour,
so it will take him 45/55 = 0.85 hour or about 51
minutes. The route on Highway 4 and Route 6 will
take less time than the route on County Line Road.
ANSWER:
a. County Line Road; sample answer: In a triangle,
the sum of two of the sides is always greater than
the third side, so the sum of the distance on Highway
4 and the distance on Route 6 is greater than the
distance on County Line Road.
b. Highway 4 to Route 6; sample answer: Since
Takoda can drive 30 miles per hour on County Line
Road and the distance is 30 miles, it will take him 1
hour. He has to drive 47 miles on Highway 4 and
Route 6, and the speed limit is 55 miles per hour, so it
will take him 0.85 hour or about 51 minutes. The
route on Highway 4 and Route 6 will take less time
than the route on County Line Road.
PROOF Write a two-column proof.
23.PROOF Write a two-column proof.
Given: ΔABC
Prove: AC + BC > AB (Triangle Inequality Theorem)
(Hint: Draw auxiliary segment , so that C is
between B and D and .)
SOLUTION:
Proof:
Statements (Reasons)
1. Construct sothatC is between B and D and
. (Ruler Postulate)
2. CD = AC (Definition of )
3. (Isosceles Theorem)
4. m CAD = m ADC (Definition. of
angles )
5. m BAC + m CAD = m BAD ( Addition
Postulate)
6. m BAC + m ADC = m BAD (Substitution)
7. m ADC < m BAD (Definition of inequality)
8. AB < BD (Angle–Side Relationships in Triangles)
9. BD = BC + CD (Segment Addition Postulate)
10. AB < BC + CD (Substitution)
11. AB < BC + AC (Substitution (Steps 2, 10))
ANSWER:
Proof:
Statements (Reasons)
1. Construct sothatC is between B and D and
.(Ruler Post.)
2. CD = AC (Def. of )
3. (Isos. Thm)
4. m CAD = m ADC (Def. of s )
5. m BAC + m CAD = m BAD ( Add.
Post.)
6. m BAC + m ADC = m BAD (Subst.)
7. m ADC < m BAD (Def. of inequality)
8. AB < BD (Angle–Side Relationships in Triangles)
9. BD = BC + CD (Seg. Add. Post.)
10. AB < BC + CD (Subst.)
11. AB < BC + AC (Subst. (Steps 2, 10))
24.SCHOOL When Toya goes from science class to
math class, she usually stops at her locker. The
distance from her science classroom to her locker is
90 feet, and the distance from her locker to her math
classroom is 110 feet. What are the possible
distances from science class to math class if she
takes the hallway that goes directly between the two
classrooms?
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than 90 +
110. Therefore, n < 90+110 or n <200.
If n is not the largest side, then 110 is the largest and
110 must be less than 90 + n. Therefore, 90 + n
>110 or n>20.
Combining these two inequalities, we get 20 < n <
200. So, the distance is greater than 20 ft and less
than 200 ft.
ANSWER:
The distance is greater than 20 ft and less than 200
ft.
Find the range of possible measures of x if each
set of expressions represents measures of the
sides of a triangle.
25.x, 4, 6
SOLUTION:
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If x is the largest side, then x must be less than 4 + 6.
Therefore, x < 4+6 or x <10.
If x is not the largest side, then 6 is the largest and 6
must be less than 4 + x. Therefore, 4 + x >6 or x >
2.
Combining these two inequalities, we get 2 < x<10.
ANSWER:
2 < x < 10
26.8, x, 12
SOLUTION:
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If x is the largest side, then x must be less than 8 +
12. Therefore, x < 8 + 12 or x < 20.
If x is not the largest side, then 12 is the largest and
12 must be less than 8 + x. Therefore, 8 + x>12 or
x > 4.
Combining these two inequalities, we get 4 < x<20.
ANSWER:
4 < x < 20
27.x + 1, 5, 7
SOLUTION:
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If x + 1 is the largest side, then x + 1 must be less
than 5 + 7. Therefore, x + 1 < 5 + 7 or x + 1 < 12 or
x <11.
If x + 1 is not the largest side, then 7 is the largest
and 7 must be less than 5 + (x + 1). Therefore, 5 + x
+ 1 > 7 or 6 + x > 7 or x > 1.
Combining these two inequalities, we get 1 < x<11.
ANSWER:
1 < x < 11
28.x – 2, 10, 12
SOLUTION:
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If x – 2 is the largest side, then x – 2 must be less
than 10 + 12. Therefore, x – 2 < 10 + 12 or x – 2 <
22 or x < 24.
If x – 2 is not the largest side, then 12 is the largest
and 12 must be less than 10 + (x – 2). Therefore, 10
+ x – 2 > 12 or 8 + x > 12 or x > 4.
Combining these two inequalities, we get 4 < x<24.
ANSWER:
4 < x < 24
29.x + 2, x + 4, x + 6
SOLUTION:
Set up and solve each of the three triangle
inequalities.
Noticethat and arealwaystruefor
any whole number measure for x. So, the only
required inequality is x > 0.
ANSWER:
x > 0
30.x, 2x + 1, x + 4
SOLUTION:
Set up and solve each of the three triangle
inequalities.
Notice that isalwaystrueforanywhole
number measure for x and isalwaystrue.So,
the required inequality is .
ANSWER:
31.Drama Club Anthony and Catherine are working on
a ramp up to the stage for the drama club's next
production. Anthony's sketch of the ramp is shown
below. Catherine is concerned about the
measurements and thinks they should recheck the
measures before they start cutting the wood. Is
Catherine'sconcernvalid?Explainyourreasoning.
SOLUTION:
Yes; sample answer: The measurements on the
drawing do not form a triangle. According to the
Triangle Inequality Theorem, the sum of the lengths
of any two sides of a triangle is greater than the
length of the third side. The lengths in the drawing
are 1 ft, ft, and ft. Since , the
triangle is impossible. They should recalculate their
measurementsbeforetheycutthewood.
ANSWER:
Yes; sample answer: The measurements on the
drawing do not form a triangle. According to the
Triangle Inequality Theorem, the sum of the lengths
of any two sides of a triangle is greater than the
length of the third side. The lengths in the drawing
are 1 ft, ft, and ft. Since , the
triangle is impossible. They should recalculate their
measurementsbeforetheycutthewood.
32.CCSS SENSE-MAKINGAisha is riding her bike
to the park and can take one of two routes. The most
direct route from her house is to take Main Street,
but it is safer to take Route 3 and then turn right on
Clay Road as shown. The additional distance she will
travel if she takes Route 3 to Clay Road is between
what two number of miles?
SOLUTION:
The distance from Aisha's house to the park via
MainSt.representsthethirdsideofatriangle.
From the Triangle Inequality Theorem the length of
this side must also be greater than 7.5 – 6 or 1.5
miles and must be less than 6 + 7.5 or 13.5 miles.
Therefore, the distance d from her house to the park
via Main St. can be represented by 1.5 < d < 13.5.
The distance to the park by taking Route 3 to Clay
Roadis7.5+6or13.5miles.
The least additional number of miles she would travel
would be greater than 13.5 –13.5 or 0. The greatest
number of additional miles she would travel would be
less than 13.5 – 1.5 or 12. Therefore, the additional
distance she will travel if she takes Route 3 to Clay
Roadisbetween0and12miles.
ANSWER:
0 and 12
33.DESIGN Carlota designed an awning that she and
her friends could take to the beach. Carlota decides
to cover the top of the awning with material that will
drape 6 inches over the front. What length of
material should she buy to use with her design so that
it covers the top of the awning, including the drape,
when the supports are open as far as possible?
Assume that the width of the material is sufficient to
cover the awning.
SOLUTION:
Let x be the length of the material needed.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If x is the largest side, then x must be less than 4 + 3.
Therefore, n < 7. Since the material will drape 6
inches or 0.5 feet over the front, the minimum length
of material she should buy is 7 + 0.5 or 7.5 feet at the
most.
ANSWER:
She should buy no more than 7.5 ft.
ESTIMATION Without using a calculator,
determine if it is possible to form a triangle with
the given side lengths. Explain.
34.
SOLUTION:
Estimate each side length by comparing the values to
perfect squares.
Since , .
is between and . Since = 1 and =
2, .
Since , .
5.9 > 2.9 + 1.5, so it is not possible to form a
triangle with the given side lengths.
ANSWER:
No; since ,
since it is between or 1 and or 2,
and since . So, .
35.
SOLUTION:
Estimate each side length by comparing the values to
perfect squares.
Since , then .
Also, since , then .
And since , then .
9.9 < 6.9 + 8.1, so yes, it is possible to form a
triangle with the given side lengths.
ANSWER:
Yes. since , since
, and since .6.9+8.1>
9.9, so it is possible.
36.
SOLUTION:
Estimate each side length by comparing the values to
perfect squares.
Since , then .
Also, since , then .
And since , then .
4.9< 3.9 + 1.9, so yes, it is possible to form a
triangle with the given side lengths.
ANSWER:
Yes. since , since
, and since .
1.9 + 3.9 > 4.9, so it is possible.
37. .
SOLUTION:
Estimate each side length by comparing the values to
perfect squares.
Since , then .
Also, since ,then .
And since , then .
11.1> 2.1 + 5.1 so no, it is not possible to form a
triangle with the given side lengths.
ANSWER:
No; since ,
since , and
since . So, 2.1 + 5.1
11.1.
CCSSREASONINGDeterminewhetherthe
given coordinates are the vertices of a triangle.
Explain.
38.X(1, –3), Y(6, 1), Z(2, 2)
SOLUTION:
We can graphically show that given coordinates form
atrianglebygraphingthem,asshownbelow.
We can algebraically prove that the given
coordinates form a triangle by proving that the
length of the longest side is greater than the sum of
thetwoshortersides.
Use the distance formula. has endpoints X(1,–3)
and Y(6, 1).
has endpoints Y(6, 1) and Z(2, 2).
has endpoints Z(2, 2) and X(1, –3).
Here, XY + YZ > XZ, XY + XZ > YZ, and XZ + YZ >
XY.
Thus,thegivencoordinatesformatriangle.
ANSWER:
Yes; XY + YZ > XZ, XY + XZ > YZ, and XZ + YZ >
XY
39.F(–4, 3), G(3, –3), H(4, 6)
SOLUTION:
We can graphically show that given coordinates form
atrianglebygraphingthem,asshownbelow.
We can algebraically prove that the given
coordinates form a triangle by proving that the
length of the longest side is greater than the sum of
thetwoshortersides.
Use the distance formula. has endpoints F(–4,3)
and G(3, 3).
has endpoints G(3, 3) and H(4, 6).
has endpoints H(4, 6) and F(–4, 3).
Here, FG + GH > FH, FG + FH > GH, and GH +
FH > FG .
Thus, the coordinated form a triangle.
ANSWER:
Yes; FG + GH > FH, FG + FH > GH, and GH +
FH > FG
40.J(–7, –1), K(9, –5), L(21, –8)
SOLUTION:
Wecangraphicallydetermineifthegiven
coordinates form a triangle by graphing them, as
shownbelow.
We can algebraically prove that the these three
points are collinear and therefore, by showing that
the sum of the two shorter segments is equal to the
longestsegment.
Use the distance formula. has endpoints J(–7,–
1) and K(9, –5).
has endpoints K(9, –5) and L(21, –8).
has endpoints L(21, –8) and J(–7, –1).
Here JK + KL = JL. You can also confirm this by
using your calculator. Compute
toconfirmthatitequals0.
Thus the given coordinates do not form a triangle.
ANSWER:
No; JK + KL = JL
41.Q(2, 6), R(6, 5), S(1, 2)
SOLUTION:
We can graphically determine if the given
coordinates form a triangle by graphing them, as
shownbelow.
We can algebraically prove that the given
coordinates form a triangle by proving that the
lengthofoneofthesidesequalszero.
Use the distance formula.
has endpoints Q(2,6) and R(6, 5).
has endpoints R(6, 5) and S(1, 2).
has endpoints S(1, 2) and Q(2, 6).
Then and
.
Thus the coordinates form a triangle.
ANSWER:
Yes;
42.MULTIPLE REPRESENTATIONS In this
problem, you will use inequalities to make
comparisons between the sides and angles of two
triangles.
a. GEOMETRIC Draw three pairs of triangles that
have two pairs of congruent sides and one pair of
sides that is not congruent. Mark each pair of
congruent sides. Label each triangle pair ABC and
DEF, where and .
b. TABULAR Copy the table below. Measure and
record the values of BC, m A, EF, and m D for
each triangle pair.
c. VERBAL Make a conjecture about the
relationship between the angles opposite the
noncongruent sides of a pair of triangles that have
two pairs of congruent legs.
SOLUTION:
a. Using a ruler, compass, or drawing tool, make sure
that and , in each of the
trianglepairsmade.
b. Use a protractor and ruler to carefully measure
the indicated lengths and angle measures in the table
below. Look for a pattern when comparing to
.
c. Sample answer: The angle opposite the longer of
the two noncongruent sides is greater than the angle
opposite the shorter of the two noncongruent sides.
ANSWER:
a.
b.
c. Sample answer: The angle opposite the longer of
the two noncongruent sides is greater than the angle
opposite the shorter of the two noncongruent sides.
43.CHALLENGE What is the range of possible
perimeters for figure ABCDE if AC = 7 and DC = 9?
Explain your reasoning.
SOLUTION:
The perimeter is greater than 36 and less than 64.
Sample answer: From the diagram we know that
and , and
because vertical angles are
congruent, so .
Using the Triangle Inequality Theorem, if 9 is the
longest length of the triangle, then the minimum
length of or is 9 – 7=2.If or is
the longest length of the triangle, then the maximum
value is 9+7=16. Therefore, the minimum value of
the total perimeter, p, of the two triangles is greater
than 2(2 + 7 + 9) or 36, and the maximum value of
the perimeter is less than 2(16 + 7 + 9) or 64 or,
expressed as an inequality, .
ANSWER:
The perimeter is greater than 36 and less than 64.
Sample answer: From the diagram we know that
and , and
because vertical angles are
congruent, so . Using the Triangle
Inequality Theorem, the minimum value of AB and
ED is 2 and the maximum value is 16. Therefore, the
minimum value of the perimeter is greater than 2(2 +
7 + 9) or 36, and the maximum value of the perimeter
is less than 2(16 + 7 + 9) or 64.
44.REASONING What is the range of lengths of each
leg of an isosceles triangle if the measure of the base
is 6 inches? Explain.
SOLUTION:
Each leg must be greater than 3 inches. According to
the Triangle Inequality Theorem, the sum of any two
sides of a triangle must be greater than the sum of
the third side. Therefore. if you consider an isosceles
triangle with lengths x, x, and 6, we know three
inequalities must hold true: .
Since the last two inequalities are
the same, we will only consider the solutions of the
firsttwo.
Since 6 >0 is always true, the solution for the lengths
of the legs of the isosceles triangle is greater than 3.
Thereisnomaximumvalue.
ANSWER:
Each leg must be greater than 3 inches. Sample
answer: When you use the Triangle Inequality
Theorem to find the minimum leg length, the solution
is greater than 3 inches. When you use it to find the
maximum leg length, the inequality is 0 < 6, which is
always true. Therefore, there is no maximum length.
45.WRITING IN MATH What can you tell about a
triangle when given three side lengths? Include at
least two items.
SOLUTION:
Sample answers: whether or not the side lengths
actually form a triangle, what the smallest and largest
angles are, whether the triangle is equilateral,
isosceles, or scalene
ANSWER:
Sample answers: whether or not the side lengths
actually form a triangle, what the smallest and largest
angles are, whether the triangle is equilateral,
isosceles, or scalene
46.CHALLENGE The sides of an isosceles triangle
are whole numbers, and its perimeter is 30 units.
What is the probability that the triangle is equilateral?
SOLUTION:
Let x be the length of the congruent sides of an
isosceles triangle. Based on the Triangle Inequality
Theorem and properties of isosceles triangles, we
know that the following inequality can be written and
solved:
Therefore, based on the given information that the
two congruent sides are whole numbers greater than
7.5 and the perimeter of the triangle is 30 units, we
can create a list of possible side lengths for this
triangle:
* 10,10 10 is equilateral so the probability of the
triangle being equilateral is .
ANSWER:
47.OPEN ENDED The length of one side of a triangle
is 2 inches. Draw a triangle in which the 2-inch side
is the shortest side and one in which the 2-inch side is
the longest side. Include side and angle measures on
your drawing.
SOLUTION:
When drawing your triangles, be sure to choose side
lengths that follow the conditions of the Triangle
Inequality Theorem. For the triangle where 2 is the
longest side length, the other two sides must each be
less than 2, however, their sum must be greater than
2. For the triangle where 2 is the shortest side, one of
the other sides plus 2 must have a greater sum than
the length of the third side. Sample sketches are
providedbelow.
ANSWER:
48.WRITING IN MATH Suppose your house is
mile from a park and the park is 1.5 miles from a
shopping center.
a. If your house, the park, and the shopping center
are noncollinear, what do you know about the
distance from your house to the shopping center?
Explain your reasoning.
b. If the three locations are collinear, what do you
know about the distance from your house to the
shopping center? Explain your reasoning.
SOLUTION:
a. Sample answer: By the Triangle Inequality
Theorem, the distance from my house to the
shopping center is greater than mile and less
than miles.
b. Sample answer: The park (P) can be between my
house (H1) and the shopping center (S), which
means that the distance from my house to the
shopping center is miles, or my house (H2) can be
between the park (P) and the shopping center (S),
which means that the distance from my house to the
shopping center is 3/4 mile.
ANSWER:
a. Sample answer: By the Triangle Inequality
Theorem, the distance from my house to the
shopping center is greater than mileandless
than miles.
b. Sample answer: The park can be between my
house and the shopping center, which means that the
distance from my house to the shopping center is
miles, or my house can be between the park and
the shopping center, which means that the distance
from my house to the shopping center is 3/4 mile.
49.If is a median of and m 1 > m 2,
which of the following statements is not true?
A AD = BD
B m ADC = m BDC
C AC > BC
D m 1 > m B
SOLUTION:
A AD = BDThisistruebecauseDisthemedian
of ,which means that D is the midpoint of .
B m ADC = m BDCThis is not true because it
is given that .
C AC > BCThis is true. Since and we
know that , then
D m 1 > m B This is true, based on the Exterior
AngleTheorem.
B is the answer.
ANSWER:
B
50.SHORT RESPONSE A high school soccer team
has a goal of winning at least 75% of their 15 games
this season. In the first three weeks, the team has
won 5 games. How many more games must the
team win to meet their goal?
SOLUTION:
75% of 15 is 11.25. The number of games should not
be in decimals, so the team has to win at least 12
games in this season. They already won 5 games, so
they must win 12 – 5 or 7 games to meet their goal.
ANSWER:
7
51.Which of the following is a logical conclusion based
on the statement and its converse below?
Statement: If a polygon is a rectangle, then it has
four sides.
Converse: If a polygon has four sides, then it is a
rectangle.
F The statement and its converse are both true.
G The statement and its converse are both
false.
H The statement is true; the converse is false.
J The statement is false; the converse is true.
SOLUTION:
The statement is correct because there exists no
contradiction. All rectangles are four-sidedpolygons.
The converse is false because there exists a
contradiction. A trapezoid is a four-sided polygon that
is not a rectangle.
Thus,Histheanswer.
ANSWER:
H
52.SAT/ACT When 7 is subtracted from 14w, the
result is z. Which of the following equations
represents this statement?
A 7 – 14w = z
B z = 14w + 7
C 7 – z = 14w
D z = 14w – 7
E 7 + 14w = 7z
SOLUTION:
A 7 – 14w = z This is not correct because 14w is
subtractedfrom7,nottheotherwayaround.
B z = 14w+7Thisisnotcorrectbecause14w and
7areadded,notsubtracted.
C 7 – z = 14wThis is not correct because the
difference of 7 and z is considered, not 14wand7.
D z = 14w – 7Thisiscorrect.
E 7 + 14w = 7z This is not correct because 14w and
7areadded,notsubtracted.
Thus, the correct answer is D.
ANSWER:
D
State the assumption you would make to start
an indirect proof of each statement.
53.If 4y + 17 = 41, then y = 6.
SOLUTION:
In an indirect proof or proof by contradiction, you
temporarily assume that what you are trying to prove
is false. By showing this assumption to be logically
impossible, you prove your assumption false and the
original conclusion true. For this problem, assume
that or .
y > 6 or y < 6
ANSWER:
y > 6 or y < 6
54.If two lines are cut by a transversal and a pair of
alternate interior angles are congruent, then the two
lines are parallel.
SOLUTION:
In an indirect proof or proof by contradiction, you
temporarily assume that what you are trying to prove
is false. By showing this assumption to be logically
impossible, you prove your assumption false and the
original conclusion true. For this problem, assume
that the two lines are not parallel..
The two lines are not parallel.
ANSWER:
The two lines are not parallel.
55.GEOGRAPHY The distance between San Jose,
California, and Las Vegas, Nevada, is about 375
miles. The distance from Las Vegas to Carlsbad,
California, is about 243 miles. Use the Triangle
Inequality Theorem to find the possible distance
between San Jose and Carlsbad.
SOLUTION:
To determine the distance between San Jose and
Carlsbad, there are two cases to consider: Case 1 is
that the three cities form a triangle with Las Vegas
andCase2isthatthethreecitiesarecollinear.
Case 1: If the three cities form a triangle, then we
can use the Triangle Inequality Theorem to find the
possiblelengthsforthethirdside.
Let d represent the distance from Carlsbad to San
Jose. Based on the Triangle Inequality Theorem, the
sum of the lengths of any two sides of a triangle must
be greater than the third side. Therefore, d + 375 >
243 or d + 243 > 375 and combining these
inequalities results in the range of values 132 < d <
618.
However, Case 2 considers that the cities may be
collinear. In this case, the distance from Carlsbad to
San Jose could be determined by the sum of the
distances from San Jose (SJ) to Las Vegas (LV) and
Las Vegas (LV) to Carlsbad (CC 1), which makes
the maximum distance 375+243= 618 miles.Or the
distance from Carlsbad to San Jose could be
determined by the difference of the distances from
San Jose (SJ) to Las Vegas (LV) and Las Vegas
(LV) to Carlsbad (CC 1), which makes the minimum
distance 375 – 243 = 132 miles.
Therefore, the answer is miles.
ANSWER:
miles
Find x so that . Identify the postulate or
theorem you used.
56.
SOLUTION:
By the Corresponding Angles Postulate,
Solve for x.
ANSWER:
16; Corr. s Post.
57.
SOLUTION:
By the Alternate Exterior Angles Theorem,
Solve for x.
ANSWER:
15; Alt. Ext. s Thm.
58.
SOLUTION:
By the Alternate Exterior Angles Theorem,
Solve for x.
ANSWER:
13; Alt. Ext. s Thm.
ALGEBRA Find x and JK if J is between K and
L.
59.KJ = 3x, JL = 6x, and KL = 12
SOLUTION:
J is between K and L. So, .
We have KJ = 3x, JL = 6x,andKL = 8.
Substitute.
Find JK.
ANSWER:
;JK = 4
60.KJ = 3x – 6, JL = x + 6, and KL = 24
SOLUTION:
J is between K and L. So, .
We have KJ = 3x – 6, JL = x+6,andKL = 24.
Substitute.
Find JK.
ANSWER:
x = 6; JK = 12
Find x and the measures of the unknown sides
of each triangle.
61.
SOLUTION:
In the figure,
So,
Solve for x.
Substitute inJK.
Since all the sides are congruent, JK = KL = LJ = 14.
ANSWER:
x = 2; JK = KL = JL = 14
62.
SOLUTION:
In the figure,
So,
Solve for x.
congruence
Triangle
congruence
Is it possible to form a triangle with the given
side lengths? If not, explain why not.
1.5 cm, 7 cm, 10 cm
SOLUTION:
The sum of the lengths of any two sides of a triangle
must be greater than the length of the third side.
Yes; 5 + 7 > 10, 5 + 10 > 7, and 7 + 10 > 5
ANSWER:
Yes; 5 + 7 > 10, 5 + 10 > 7, and 7 + 10 > 5
2.3 in., 4 in., 8 in.
SOLUTION:
No; . The sum of the lengths of any two
sides of a triangle must be greater than the length of
the third side.
ANSWER:
No;
3.6 m, 14 m, 10 m
SOLUTION:
Yes;6+14>10,6+10>14,and10+14>6.
The sum of the lengths of any two sides of a triangle
must be greater than the length of the third side.
ANSWER:
Yes; 6 + 14 > 10, 6 + 10 > 14, and 10 + 14 > 6
4.MULTIPLE CHOICE If the measures of two
sides of a triangle are 5 yards and 9 yards, what is
the least possible measure of the third side if the
measure is an integer?
A 4 yd
B 5 yd
C 6 yd
D 14 yd
SOLUTION:
Let x represents the length of the third side. Next, set
up and solve each of the three triangle inequalities.
5 + 9 > x, 5 + x > 9, and 9 + x > 5
That is, 14 > x, x > 4, and x > –4.
Notice that x > –4 is always true for any whole
number measure for x. Combining the two remaining
inequalities, the range of values that fit both
inequalities is x > 4 and x < 14, which can be written
as 4 < x < 14. So, the least possible measure of the
thirdsidecouldbe5yd.
The correct option is B.
ANSWER:
B
PROOF Write a two-column proof.
5.Given:
Prove:
SOLUTION:
Think backwards when considering this proof. Notice
that what you are trying to prove is an inequality
statement. However, it isn't exactly related to
, except for instead of side being used, it
is Since it is given that , you can
easilyusethisinasubstitutionstep.
Given:
Prove: YZ + ZW > XW
Statements (Reasons)
1. (Given)
2. XW = YW (Def. of segments)
3. YZ + ZW > YW ( Inequal. Thm.)
4. YZ + ZW > XW (Substitution Property.)
ANSWER:
Given:
Prove: YZ + ZW > XW
Statements (Reasons)
1. (Given)
2. XW = YW (Def. of segments)
3. YZ + ZW > YW ( Inequal. Thm.)
4. YZ + ZW > XW (Subst.)
Is it possible to form a triangle with the given
side lengths? If not, explain why not.
6.4 ft, 9 ft, 15 ft
SOLUTION:
No; . The sum of the lengths of any two
sides of a triangle must be greater than the length of
the third side.
ANSWER:
No;
7.11 mm, 21 mm, 16 mm
SOLUTION:
Yes;11+21>16,11+16>21,and16+21>11.
The sum of the lengths of any two sides of a triangle
must be greater than the length of the third side.
ANSWER:
Yes; 11 + 21 > 16, 11 + 16 > 21, and 16 + 21 > 11
8.9.9 cm, 1.1 cm, 8.2 cm
SOLUTION:
No; The sum of the lengths of any two
sides of a triangle must be greater than the length of
the third side.
ANSWER:
No;
9.2.1 in., 4.2 in., 7.9 in.
SOLUTION:
No; The sum of the lengths of any
two sides of a triangle must be greater than the
length of the third side.
ANSWER:
No;
10.
SOLUTION:
No; The sum of the lengths of any two
sides of a triangle must be greater than the length of
the third side.
ANSWER:
No;
11.
SOLUTION:
Yes;
The sum of the lengths of any two sides of a triangle
must be greater than the length of the third side.
ANSWER:
Yes;
Find the range for the measure of the third side
of a triangle given the measures of two sides.
12.4 ft, 8 ft
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than 4 +
8. Therefore, n < 12.
If n is not the largest side, then 8 is the largest and 8
must be less than 4 + n. Therefore, 4 < n.
Combining these two inequalities, we get 4 < n < 12.
ANSWER:
4 ft < n < 12 ft
13.5 m, 11 m
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than 5
+11. Therefore, n < 16.
If n is not the largest side, then 11 is the largest and
11 must be less than 5 + n. Therefore, 6 < n.
Combining these two inequalities, we get 6 < n < 16.
ANSWER:
6 m < n < 16 m
14.2.7 cm, 4.2 cm
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than 2.7 +
4.2. Therefore, n < 6.9.
If n is not the largest side, then 4.2 is the largest and
4.2 must be less than 2.7 + n. Therefore, 1.5 < n.
Combining these two inequalities, we get 1.5 < n <
6.9.
ANSWER:
1.5 cm < n < 6.9 cm
15.3.8 in., 9.2 in.
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than 3.8 +
9.2. Therefore, n < 13.
If n is not the largest side, then 9.2 is the largest and
9.2 must be less than 3.8 + n. Therefore, 5.4 < n.
Combining these two inequalities, we get 5.4 < n <
13.
ANSWER:
5.4 in. < n < 13 in.
16.
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than
. Therefore, .
If n is not the largest side, then is the largest and
must be less than . Therefore, .
Combining these two inequalities, we get
.
ANSWER:
17.
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than
. Therefore, n < .
If n is not the largest side, then is the largest and
must be less than . Therefore, .
Combining these two inequalities, we get
.
ANSWER:
PROOF Write a two-column proof.
18.Given:
Prove:
SOLUTION:
The key to this proof is to figure out some way to get
BC=BD so that you can substitute one in for the
other using the Triangle Inequality Theorem.
Consider the given statement, if two angles of a
triangle are congruent, what kind of triangle is it and ,
therefore, how do you know that BC must equal BD?
Proof:
Statements (Reasons)
1. (Given)
2. (Converse of Isosceles Thm.)
3. BC = BD (Def. of segments)
4. AB + AD > BD ( Inequality Thm.)
5. AB + AD > BC (Substitution Property.)
ANSWER:
Proof:
Statements (Reasons)
1. (Given)
2. (Conv.Isos. Thm.)
3. BC = BD (Def. of segments)
4. AB + AD > BD ( Inequal. Thm.)
5. AB + AD > BC (Subst.)
19.Given:
Prove: KJ +KL> LM
SOLUTION:
Think backwards when considering this proof. Notice
that what you are trying to prove is an inequality
statement. However, it isn't exactly related to
, except for instead of side being used, it
is Since it is given that , you can
easily use this in a substitution step
Proof:
Statements (Reasons)
1. (Given)
2. JL = LM (Def. of segments)
3. KJ + KL > JL ( Inequality Thm.)
4. KJ + KL > LM (Substitution Property)
ANSWER:
Proof:
Statements (Reasons)
1. (Given)
2. JL = LM (Def. of segments)
3. KJ + KL > JL ( Inequal. Thm.)
4. KJ + KL > LM (Subst.)
CCSS SENSE-MAKINGDetermine the
possible values of x.
20.
SOLUTION:
Set up and solve each of the three triangle
inequalities.
Notice that isalwaystrueforanywhole
number measure for x.The range of values that
would be true for the other two inequalities is
and , which can be written as
.
ANSWER:
6 < x < 17
21.
SOLUTION:
Set up and solve each of the three triangle
inequalities.
Notice that isalwaystrueforanywhole
number measure for x. Combining the two remaining
inequalities, the range of values that fit both
inequalities is and , which can be
written as .
ANSWER:
22.DRIVING Takoda wants to take the most efficient
route from his house to a soccer tournament at The
Sportsplex. He can take County Line Road or he can
take Highway 4 and then Route 6 to the get to The
Sportsplex.
a. Which of the two possible routes is the shortest?
Explain your reasoning.
b. Suppose Takoda always drives below the speed
limit. If the speed limit on County Line Road is 30
miles per hour and on both Highway 4 and Route 6 it
is 55 miles per hour, which route will be faster?
Explain.
SOLUTION:
a. County Line Road; sample answer: In a triangle,
the sum of two of the sides is always greater than
the third side, so the sum of the distance on Highway
4 and the distance on Route 6 is greater than the
distance on County Line Road. Or you can add the
distances using Highway 4 and Route 6 and compare
their sum to the 30 miles of County Line Road. Since
47 miles is greater than 30 miles, County Line Road
istheshortestdistance.
b. Highway 4 to Route 6; sample answer: Since
Takoda drives below the 30 mph speed limit on
County Line Road and the distance is 30 miles, it will
take him about 30/30 = 1 hour to get to The
Sportsplex. He has to drive 47 miles on Highway 4
and Route 6, and the speed limit is 55 miles per hour,
so it will take him 45/55 = 0.85 hour or about 51
minutes. The route on Highway 4 and Route 6 will
take less time than the route on County Line Road.
ANSWER:
a. County Line Road; sample answer: In a triangle,
the sum of two of the sides is always greater than
the third side, so the sum of the distance on Highway
4 and the distance on Route 6 is greater than the
distance on County Line Road.
b. Highway 4 to Route 6; sample answer: Since
Takoda can drive 30 miles per hour on County Line
Road and the distance is 30 miles, it will take him 1
hour. He has to drive 47 miles on Highway 4 and
Route 6, and the speed limit is 55 miles per hour, so it
will take him 0.85 hour or about 51 minutes. The
route on Highway 4 and Route 6 will take less time
than the route on County Line Road.
PROOF Write a two-column proof.
23.PROOF Write a two-column proof.
Given: ΔABC
Prove: AC + BC > AB (Triangle Inequality Theorem)
(Hint: Draw auxiliary segment , so that C is
between B and D and .)
SOLUTION:
Proof:
Statements (Reasons)
1. Construct sothatC is between B and D and
. (Ruler Postulate)
2. CD = AC (Definition of )
3. (Isosceles Theorem)
4. m CAD = m ADC (Definition. of
angles )
5. m BAC + m CAD = m BAD ( Addition
Postulate)
6. m BAC + m ADC = m BAD (Substitution)
7. m ADC < m BAD (Definition of inequality)
8. AB < BD (Angle–Side Relationships in Triangles)
9. BD = BC + CD (Segment Addition Postulate)
10. AB < BC + CD (Substitution)
11. AB < BC + AC (Substitution (Steps 2, 10))
ANSWER:
Proof:
Statements (Reasons)
1. Construct sothatC is between B and D and
.(Ruler Post.)
2. CD = AC (Def. of )
3. (Isos. Thm)
4. m CAD = m ADC (Def. of s )
5. m BAC + m CAD = m BAD ( Add.
Post.)
6. m BAC + m ADC = m BAD (Subst.)
7. m ADC < m BAD (Def. of inequality)
8. AB < BD (Angle–Side Relationships in Triangles)
9. BD = BC + CD (Seg. Add. Post.)
10. AB < BC + CD (Subst.)
11. AB < BC + AC (Subst. (Steps 2, 10))
24.SCHOOL When Toya goes from science class to
math class, she usually stops at her locker. The
distance from her science classroom to her locker is
90 feet, and the distance from her locker to her math
classroom is 110 feet. What are the possible
distances from science class to math class if she
takes the hallway that goes directly between the two
classrooms?
SOLUTION:
Let nrepresentthelengthofthethirdside.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If n is the largest side, then n must be less than 90 +
110. Therefore, n < 90+110 or n <200.
If n is not the largest side, then 110 is the largest and
110 must be less than 90 + n. Therefore, 90 + n
>110 or n>20.
Combining these two inequalities, we get 20 < n <
200. So, the distance is greater than 20 ft and less
than 200 ft.
ANSWER:
The distance is greater than 20 ft and less than 200
ft.
Find the range of possible measures of x if each
set of expressions represents measures of the
sides of a triangle.
25.x, 4, 6
SOLUTION:
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If x is the largest side, then x must be less than 4 + 6.
Therefore, x < 4+6 or x <10.
If x is not the largest side, then 6 is the largest and 6
must be less than 4 + x. Therefore, 4 + x >6 or x >
2.
Combining these two inequalities, we get 2 < x<10.
ANSWER:
2 < x < 10
26.8, x, 12
SOLUTION:
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If x is the largest side, then x must be less than 8 +
12. Therefore, x < 8 + 12 or x < 20.
If x is not the largest side, then 12 is the largest and
12 must be less than 8 + x. Therefore, 8 + x>12 or
x > 4.
Combining these two inequalities, we get 4 < x<20.
ANSWER:
4 < x < 20
27.x + 1, 5, 7
SOLUTION:
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If x + 1 is the largest side, then x + 1 must be less
than 5 + 7. Therefore, x + 1 < 5 + 7 or x + 1 < 12 or
x <11.
If x + 1 is not the largest side, then 7 is the largest
and 7 must be less than 5 + (x + 1). Therefore, 5 + x
+ 1 > 7 or 6 + x > 7 or x > 1.
Combining these two inequalities, we get 1 < x<11.
ANSWER:
1 < x < 11
28.x – 2, 10, 12
SOLUTION:
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If x – 2 is the largest side, then x – 2 must be less
than 10 + 12. Therefore, x – 2 < 10 + 12 or x – 2 <
22 or x < 24.
If x – 2 is not the largest side, then 12 is the largest
and 12 must be less than 10 + (x – 2). Therefore, 10
+ x – 2 > 12 or 8 + x > 12 or x > 4.
Combining these two inequalities, we get 4 < x<24.
ANSWER:
4 < x < 24
29.x + 2, x + 4, x + 6
SOLUTION:
Set up and solve each of the three triangle
inequalities.
Noticethat and arealwaystruefor
any whole number measure for x. So, the only
required inequality is x > 0.
ANSWER:
x > 0
30.x, 2x + 1, x + 4
SOLUTION:
Set up and solve each of the three triangle
inequalities.
Notice that isalwaystrueforanywhole
number measure for x and isalwaystrue.So,
the required inequality is .
ANSWER:
31.Drama Club Anthony and Catherine are working on
a ramp up to the stage for the drama club's next
production. Anthony's sketch of the ramp is shown
below. Catherine is concerned about the
measurements and thinks they should recheck the
measures before they start cutting the wood. Is
Catherine'sconcernvalid?Explainyourreasoning.
SOLUTION:
Yes; sample answer: The measurements on the
drawing do not form a triangle. According to the
Triangle Inequality Theorem, the sum of the lengths
of any two sides of a triangle is greater than the
length of the third side. The lengths in the drawing
are 1 ft, ft, and ft. Since , the
triangle is impossible. They should recalculate their
measurementsbeforetheycutthewood.
ANSWER:
Yes; sample answer: The measurements on the
drawing do not form a triangle. According to the
Triangle Inequality Theorem, the sum of the lengths
of any two sides of a triangle is greater than the
length of the third side. The lengths in the drawing
are 1 ft, ft, and ft. Since , the
triangle is impossible. They should recalculate their
measurementsbeforetheycutthewood.
32.CCSS SENSE-MAKINGAisha is riding her bike
to the park and can take one of two routes. The most
direct route from her house is to take Main Street,
but it is safer to take Route 3 and then turn right on
Clay Road as shown. The additional distance she will
travel if she takes Route 3 to Clay Road is between
what two number of miles?
SOLUTION:
The distance from Aisha's house to the park via
MainSt.representsthethirdsideofatriangle.
From the Triangle Inequality Theorem the length of
this side must also be greater than 7.5 – 6 or 1.5
miles and must be less than 6 + 7.5 or 13.5 miles.
Therefore, the distance d from her house to the park
via Main St. can be represented by 1.5 < d < 13.5.
The distance to the park by taking Route 3 to Clay
Roadis7.5+6or13.5miles.
The least additional number of miles she would travel
would be greater than 13.5 –13.5 or 0. The greatest
number of additional miles she would travel would be
less than 13.5 – 1.5 or 12. Therefore, the additional
distance she will travel if she takes Route 3 to Clay
Roadisbetween0and12miles.
ANSWER:
0 and 12
33.DESIGN Carlota designed an awning that she and
her friends could take to the beach. Carlota decides
to cover the top of the awning with material that will
drape 6 inches over the front. What length of
material should she buy to use with her design so that
it covers the top of the awning, including the drape,
when the supports are open as far as possible?
Assume that the width of the material is sufficient to
cover the awning.
SOLUTION:
Let x be the length of the material needed.
According to the Triangle Inequality Theorem, the
largest side cannot be greater than the sum of the
othertwosides.
If x is the largest side, then x must be less than 4 + 3.
Therefore, n < 7. Since the material will drape 6
inches or 0.5 feet over the front, the minimum length
of material she should buy is 7 + 0.5 or 7.5 feet at the
most.
ANSWER:
She should buy no more than 7.5 ft.
ESTIMATION Without using a calculator,
determine if it is possible to form a triangle with
the given side lengths. Explain.
34.
SOLUTION:
Estimate each side length by comparing the values to
perfect squares.
Since , .
is between and . Since = 1 and =
2, .
Since , .
5.9 > 2.9 + 1.5, so it is not possible to form a
triangle with the given side lengths.
ANSWER:
No; since ,
since it is between or 1 and or 2,
and since . So, .
35.
SOLUTION:
Estimate each side length by comparing the values to
perfect squares.
Since , then .
Also, since , then .
And since , then .
9.9 < 6.9 + 8.1, so yes, it is possible to form a
triangle with the given side lengths.
ANSWER:
Yes. since , since
, and since .6.9+8.1>
9.9, so it is possible.
36.
SOLUTION:
Estimate each side length by comparing the values to
perfect squares.
Since , then .
Also, since , then .
And since , then .
4.9< 3.9 + 1.9, so yes, it is possible to form a
triangle with the given side lengths.
ANSWER:
Yes. since , since
, and since .
1.9 + 3.9 > 4.9, so it is possible.
37. .
SOLUTION:
Estimate each side length by comparing the values to
perfect squares.
Since , then .
Also, since ,then .
And since , then .
11.1> 2.1 + 5.1 so no, it is not possible to form a
triangle with the given side lengths.
ANSWER:
No; since ,
since , and
since . So, 2.1 + 5.1
11.1.
CCSSREASONINGDeterminewhetherthe
given coordinates are the vertices of a triangle.
Explain.
38.X(1, –3), Y(6, 1), Z(2, 2)
SOLUTION:
We can graphically show that given coordinates form
atrianglebygraphingthem,asshownbelow.
We can algebraically prove that the given
coordinates form a triangle by proving that the
length of the longest side is greater than the sum of
thetwoshortersides.
Use the distance formula. has endpoints X(1,–3)
and Y(6, 1).
has endpoints Y(6, 1) and Z(2, 2).
has endpoints Z(2, 2) and X(1, –3).
Here, XY + YZ > XZ, XY + XZ > YZ, and XZ + YZ >
XY.
Thus,thegivencoordinatesformatriangle.
ANSWER:
Yes; XY + YZ > XZ, XY + XZ > YZ, and XZ + YZ >
XY
39.F(–4, 3), G(3, –3), H(4, 6)
SOLUTION:
We can graphically show that given coordinates form
atrianglebygraphingthem,asshownbelow.
We can algebraically prove that the given
coordinates form a triangle by proving that the
length of the longest side is greater than the sum of
thetwoshortersides.
Use the distance formula. has endpoints F(–4,3)
and G(3, 3).
has endpoints G(3, 3) and H(4, 6).
has endpoints H(4, 6) and F(–4, 3).
Here, FG + GH > FH, FG + FH > GH, and GH +
FH > FG .
Thus, the coordinated form a triangle.
ANSWER:
Yes; FG + GH > FH, FG + FH > GH, and GH +
FH > FG
40.J(–7, –1), K(9, –5), L(21, –8)
SOLUTION:
Wecangraphicallydetermineifthegiven
coordinates form a triangle by graphing them, as
shownbelow.
We can algebraically prove that the these three
points are collinear and therefore, by showing that
the sum of the two shorter segments is equal to the
longestsegment.
Use the distance formula. has endpoints J(–7,–
1) and K(9, –5).
has endpoints K(9, –5) and L(21, –8).
has endpoints L(21, –8) and J(–7, –1).
Here JK + KL = JL. You can also confirm this by
using your calculator. Compute
toconfirmthatitequals0.
Thus the given coordinates do not form a triangle.
ANSWER:
No; JK + KL = JL
41.Q(2, 6), R(6, 5), S(1, 2)
SOLUTION:
We can graphically determine if the given
coordinates form a triangle by graphing them, as
shownbelow.
We can algebraically prove that the given
coordinates form a triangle by proving that the
lengthofoneofthesidesequalszero.
Use the distance formula.
has endpoints Q(2,6) and R(6, 5).
has endpoints R(6, 5) and S(1, 2).
has endpoints S(1, 2) and Q(2, 6).
Then and
.
Thus the coordinates form a triangle.
ANSWER:
Yes;
42.MULTIPLE REPRESENTATIONS In this
problem, you will use inequalities to make
comparisons between the sides and angles of two
triangles.
a. GEOMETRIC Draw three pairs of triangles that
have two pairs of congruent sides and one pair of
sides that is not congruent. Mark each pair of
congruent sides. Label each triangle pair ABC and
DEF, where and .
b. TABULAR Copy the table below. Measure and
record the values of BC, m A, EF, and m D for
each triangle pair.
c. VERBAL Make a conjecture about the
relationship between the angles opposite the
noncongruent sides of a pair of triangles that have
two pairs of congruent legs.
SOLUTION:
a. Using a ruler, compass, or drawing tool, make sure
that and , in each of the
trianglepairsmade.
b. Use a protractor and ruler to carefully measure
the indicated lengths and angle measures in the table
below. Look for a pattern when comparing to
.
c. Sample answer: The angle opposite the longer of
the two noncongruent sides is greater than the angle
opposite the shorter of the two noncongruent sides.
ANSWER:
a.
b.
c. Sample answer: The angle opposite the longer of
the two noncongruent sides is greater than the angle
opposite the shorter of the two noncongruent sides.
43.CHALLENGE What is the range of possible
perimeters for figure ABCDE if AC = 7 and DC = 9?
Explain your reasoning.
SOLUTION:
The perimeter is greater than 36 and less than 64.
Sample answer: From the diagram we know that
and , and
because vertical angles are
congruent, so .
Using the Triangle Inequality Theorem, if 9 is the
longest length of the triangle, then the minimum
length of or is 9 – 7=2.If or is
the longest length of the triangle, then the maximum
value is 9+7=16. Therefore, the minimum value of
the total perimeter, p, of the two triangles is greater
than 2(2 + 7 + 9) or 36, and the maximum value of
the perimeter is less than 2(16 + 7 + 9) or 64 or,
expressed as an inequality, .
ANSWER:
The perimeter is greater than 36 and less than 64.
Sample answer: From the diagram we know that
and , and
because vertical angles are
congruent, so . Using the Triangle
Inequality Theorem, the minimum value of AB and
ED is 2 and the maximum value is 16. Therefore, the
minimum value of the perimeter is greater than 2(2 +
7 + 9) or 36, and the maximum value of the perimeter
is less than 2(16 + 7 + 9) or 64.
44.REASONING What is the range of lengths of each
leg of an isosceles triangle if the measure of the base
is 6 inches? Explain.
SOLUTION:
Each leg must be greater than 3 inches. According to
the Triangle Inequality Theorem, the sum of any two
sides of a triangle must be greater than the sum of
the third side. Therefore. if you consider an isosceles
triangle with lengths x, x, and 6, we know three
inequalities must hold true: .
Since the last two inequalities are
the same, we will only consider the solutions of the
firsttwo.
Since 6 >0 is always true, the solution for the lengths
of the legs of the isosceles triangle is greater than 3.
Thereisnomaximumvalue.
ANSWER:
Each leg must be greater than 3 inches. Sample
answer: When you use the Triangle Inequality
Theorem to find the minimum leg length, the solution
is greater than 3 inches. When you use it to find the
maximum leg length, the inequality is 0 < 6, which is
always true. Therefore, there is no maximum length.
45.WRITING IN MATH What can you tell about a
triangle when given three side lengths? Include at
least two items.
SOLUTION:
Sample answers: whether or not the side lengths
actually form a triangle, what the smallest and largest
angles are, whether the triangle is equilateral,
isosceles, or scalene
ANSWER:
Sample answers: whether or not the side lengths
actually form a triangle, what the smallest and largest
angles are, whether the triangle is equilateral,
isosceles, or scalene
46.CHALLENGE The sides of an isosceles triangle
are whole numbers, and its perimeter is 30 units.
What is the probability that the triangle is equilateral?
SOLUTION:
Let x be the length of the congruent sides of an
isosceles triangle. Based on the Triangle Inequality
Theorem and properties of isosceles triangles, we
know that the following inequality can be written and
solved:
Therefore, based on the given information that the
two congruent sides are whole numbers greater than
7.5 and the perimeter of the triangle is 30 units, we
can create a list of possible side lengths for this
triangle:
* 10,10 10 is equilateral so the probability of the
triangle being equilateral is .
ANSWER:
47.OPEN ENDED The length of one side of a triangle
is 2 inches. Draw a triangle in which the 2-inch side
is the shortest side and one in which the 2-inch side is
the longest side. Include side and angle measures on
your drawing.
SOLUTION:
When drawing your triangles, be sure to choose side
lengths that follow the conditions of the Triangle
Inequality Theorem. For the triangle where 2 is the
longest side length, the other two sides must each be
less than 2, however, their sum must be greater than
2. For the triangle where 2 is the shortest side, one of
the other sides plus 2 must have a greater sum than
the length of the third side. Sample sketches are
providedbelow.
ANSWER:
48.WRITING IN MATH Suppose your house is
mile from a park and the park is 1.5 miles from a
shopping center.
a. If your house, the park, and the shopping center
are noncollinear, what do you know about the
distance from your house to the shopping center?
Explain your reasoning.
b. If the three locations are collinear, what do you
know about the distance from your house to the
shopping center? Explain your reasoning.
SOLUTION:
a. Sample answer: By the Triangle Inequality
Theorem, the distance from my house to the
shopping center is greater than mile and less
than miles.
b. Sample answer: The park (P) can be between my
house (H1) and the shopping center (S), which
means that the distance from my house to the
shopping center is miles, or my house (H2) can be
between the park (P) and the shopping center (S),
which means that the distance from my house to the
shopping center is 3/4 mile.
ANSWER:
a. Sample answer: By the Triangle Inequality
Theorem, the distance from my house to the
shopping center is greater than mileandless
than miles.
b. Sample answer: The park can be between my
house and the shopping center, which means that the
distance from my house to the shopping center is
miles, or my house can be between the park and
the shopping center, which means that the distance
from my house to the shopping center is 3/4 mile.
49.If is a median of and m 1 > m 2,
which of the following statements is not true?
A AD = BD
B m ADC = m BDC
C AC > BC
D m 1 > m B
SOLUTION:
A AD = BDThisistruebecauseDisthemedian
of ,which means that D is the midpoint of .
B m ADC = m BDCThis is not true because it
is given that .
C AC > BCThis is true. Since and we
know that , then
D m 1 > m B This is true, based on the Exterior
AngleTheorem.
B is the answer.
ANSWER:
B
50.SHORT RESPONSE A high school soccer team
has a goal of winning at least 75% of their 15 games
this season. In the first three weeks, the team has
won 5 games. How many more games must the
team win to meet their goal?
SOLUTION:
75% of 15 is 11.25. The number of games should not
be in decimals, so the team has to win at least 12
games in this season. They already won 5 games, so
they must win 12 – 5 or 7 games to meet their goal.
ANSWER:
7
51.Which of the following is a logical conclusion based
on the statement and its converse below?
Statement: If a polygon is a rectangle, then it has
four sides.
Converse: If a polygon has four sides, then it is a
rectangle.
F The statement and its converse are both true.
G The statement and its converse are both
false.
H The statement is true; the converse is false.
J The statement is false; the converse is true.
SOLUTION:
The statement is correct because there exists no
contradiction. All rectangles are four-sidedpolygons.
The converse is false because there exists a
contradiction. A trapezoid is a four-sided polygon that
is not a rectangle.
Thus,Histheanswer.
ANSWER:
H
52.SAT/ACT When 7 is subtracted from 14w, the
result is z. Which of the following equations
represents this statement?
A 7 – 14w = z
B z = 14w + 7
C 7 – z = 14w
D z = 14w – 7
E 7 + 14w = 7z
SOLUTION:
A 7 – 14w = z This is not correct because 14w is
subtractedfrom7,nottheotherwayaround.
B z = 14w+7Thisisnotcorrectbecause14w and
7areadded,notsubtracted.
C 7 – z = 14wThis is not correct because the
difference of 7 and z is considered, not 14wand7.
D z = 14w – 7Thisiscorrect.
E 7 + 14w = 7z This is not correct because 14w and
7areadded,notsubtracted.
Thus, the correct answer is D.
ANSWER:
D
State the assumption you would make to start
an indirect proof of each statement.
53.If 4y + 17 = 41, then y = 6.
SOLUTION:
In an indirect proof or proof by contradiction, you
temporarily assume that what you are trying to prove
is false. By showing this assumption to be logically
impossible, you prove your assumption false and the
original conclusion true. For this problem, assume
that or .
y > 6 or y < 6
ANSWER:
y > 6 or y < 6
54.If two lines are cut by a transversal and a pair of
alternate interior angles are congruent, then the two
lines are parallel.
SOLUTION:
In an indirect proof or proof by contradiction, you
temporarily assume that what you are trying to prove
is false. By showing this assumption to be logically
impossible, you prove your assumption false and the
original conclusion true. For this problem, assume
that the two lines are not parallel..
The two lines are not parallel.
ANSWER:
The two lines are not parallel.
55.GEOGRAPHY The distance between San Jose,
California, and Las Vegas, Nevada, is about 375
miles. The distance from Las Vegas to Carlsbad,
California, is about 243 miles. Use the Triangle
Inequality Theorem to find the possible distance
between San Jose and Carlsbad.
SOLUTION:
To determine the distance between San Jose and
Carlsbad, there are two cases to consider: Case 1 is
that the three cities form a triangle with Las Vegas
andCase2isthatthethreecitiesarecollinear.
Case 1: If the three cities form a triangle, then we
can use the Triangle Inequality Theorem to find the
possiblelengthsforthethirdside.
Let d represent the distance from Carlsbad to San
Jose. Based on the Triangle Inequality Theorem, the
sum of the lengths of any two sides of a triangle must
be greater than the third side. Therefore, d + 375 >
243 or d + 243 > 375 and combining these
inequalities results in the range of values 132 < d <
618.
However, Case 2 considers that the cities may be
collinear. In this case, the distance from Carlsbad to
San Jose could be determined by the sum of the
distances from San Jose (SJ) to Las Vegas (LV) and
Las Vegas (LV) to Carlsbad (CC 1), which makes
the maximum distance 375+243= 618 miles.Or the
distance from Carlsbad to San Jose could be
determined by the difference of the distances from
San Jose (SJ) to Las Vegas (LV) and Las Vegas
(LV) to Carlsbad (CC 1), which makes the minimum
distance 375 – 243 = 132 miles.
Therefore, the answer is miles.
ANSWER:
miles
Find x so that . Identify the postulate or
theorem you used.
56.
SOLUTION:
By the Corresponding Angles Postulate,
Solve for x.
ANSWER:
16; Corr. s Post.
57.
SOLUTION:
By the Alternate Exterior Angles Theorem,
Solve for x.
ANSWER:
15; Alt. Ext. s Thm.
58.
SOLUTION:
By the Alternate Exterior Angles Theorem,
Solve for x.
ANSWER:
13; Alt. Ext. s Thm.
ALGEBRA Find x and JK if J is between K and
L.
59.KJ = 3x, JL = 6x, and KL = 12
SOLUTION:
J is between K and L. So, .
We have KJ = 3x, JL = 6x,andKL = 8.
Substitute.
Find JK.
ANSWER:
;JK = 4
60.KJ = 3x – 6, JL = x + 6, and KL = 24
SOLUTION:
J is between K and L. So, .
We have KJ = 3x – 6, JL = x+6,andKL = 24.
Substitute.
Find JK.
ANSWER:
x = 6; JK = 12
Find x and the measures of the unknown sides
of each triangle.
61.
SOLUTION:
In the figure,
So,
Solve for x.
Substitute inJK.
Since all the sides are congruent, JK = KL = LJ = 14.
ANSWER:
x = 2; JK = KL = JL = 14
62.
SOLUTION:
In the figure,
So,
Solve for x.
congruence
Triangle
congruence
eSolutionsManual-PoweredbyCogneroPage1
5-5 The Triangle Inequality
