1.CCSSARGUMENTSCopy and complete the
proof.
Given:
Prove:
Proof:
SOLUTION:
The 1st row is the information given.
The 2nd row the definition of congruent segment,
changes congruent symbols changed to equal signs
and the remove the bars above the segment to
indicate the lengths of the segments.
The 3rd row is found adding the two congruent
segments.
The 4th row is uses the Segment Addition Postulate
to rewrite the segments.
The 5th row is substitution to replace the segments
with equivalent segments.
The 6th row is replacing the = with congruent
symbols and changing the segments lengths to
segment using the definition of congruent segments. .
2.PROOF Prove the following.
Given:
Prove:
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent segments.
You need to find a way to relate the smaller segment
with the larger segments. Use the properties that you
have learned about congruent segments and
equivalent expressions in algebra to walk through the
proof.
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. WX =YZ ( Definition of congruent segments)
3. XY = XY (Reflexive Property)
4. WX + XY = XY + YZ (Addition Property)
5. WY = WX + XY; XZ = XY + YZ (Segment Addition
Postulate)
6. WY = XZ (Substitution.)
7. (Definitionofcongruentsegments)
3.SCISSORS Refer to the diagram shown. is
congruent to . iscongruentto . Prove
that AP + DP = CP + BP.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two pairs of congruent
segments. Use the properties that you have learned
about congruent segments and equivalent expressions
in algebra to walk through the proof.
Given:
Prove: AR + DP = CP + BP
Proof:
Statements (Reasons)
1. (Given)
2. AR = CP, = DP = BP(Definitionofcongruent
segments)
3. AR + DP = CP + DP(AdditionProperty)
4. AR + DP = CP + BP(Substitution.)
4.CCSS ARGUMENTS Copy and complete the
proof.
Given: C is the midpoint of .
C is the midpoint of .
Prove:
Proof:
SOLUTION:
The 1st contains the given information about the
midpoints of segments and segment congruence.
The 2nd row is uses the midpoint of each segment to
write equivalence using the definition of midpoints.
The 3rd row changes the segment congruence to
distance equivalence using the definition of congruent
segments.
The4throwrewritestwoequalsegmentseachwith
two parts.
The 5th row is replaces or substitutes the two equal
segments with their parts.
The 6th row is replaces or substitutes segments with
congruent segments.
The 7th row is simplifying or replacing segments by
combining them.
The 8th row is found by dividing by 2.
The 9th row is changing segment length to segment
congruence using definition of congruent segments.
5.TILING A tile setter cuts a piece of tile to a desired
length. He then uses this tile as a pattern to cut a
second tile congruent to the first. He uses the first
two tiles to cut a third tile whose length is the sum of
the measures of the first two tiles. Prove that the
measure of the third tile is twice the measure of the
first tile.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent tiles and
another tile which is equal in length to the sum of the
lengths of the other tiles. You need to find a way to
relate the large tile with the first small tile. Use the
properties that you have learned about congruent
segments and equivalent expressions in algebra to
walk through the proof.
Given: , AB + CD = EF
Prove: 2AB = EF
Statements (Reasons)
1. , AB + CD = EF (Given)
2. AB = CD (Definition of congruent segment)
3. AB + AB = EF (Substitution)
4. 2AB = EF (Substitution Property)
CCSSARGUMENTSProveeachtheorem.
6.Symmetric Property of Congruence (Theorem 2.2)
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent congruent
segment. You need to find a way to relate the
congruent segments to itself in different order. Use
the properties that you have learned about congruent
segments and equivalent expressions in algebra to
walk through the proof.
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. AB = CD (Definition of congruent segments)
3. CD = AB (Symmetric. Properties)
4. (Definitionofcongruentsegments)
7.Reflexive Property of Congruence (Theorem 2.2)
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are segment. You need to find a
way to relate the the segment to itself. Use the
properties that you have learned about congruent
segments and equivalent expressions in algebra to
walk through the proof.
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. AB = AB (Reflexive. Property)
3. (Definitionofcongruentsegments.)
8.TRAVEL Four cities in New York are connected
by Interstate 90: Buffalo, Utica, Albany, and
Syracuse. Buffalo is the farthest west.
•Albany is 126 miles from Syracuse and 263 miles
from Buffalo.
•Buffalo is 137 miles from Syracuse and 184 miles
from Utica.
a. Draw a diagram to represent the locations of the
cities in relation to each other and the distances
between each city. Assume that Interstate 90 is
straight.
b. Write a paragraph proof to support your
conclusion.
SOLUTION:
a.Use the information given to determine the order
of the cities. Since Buffalo is furthest west, start with
the information given about Buffalo. Albany is 263
miles from Buffalo. Buffalo is 137 miles from
Syracuse. So Syracuse must be between Buffalo and
Albany. Next, Buffalo is 184 miles from Utica. So,
Utica must be between Syracuse and Albany.
Next, find the distances between each consecutive
pair of cities. The distance from Syracuse to Utica is
184 - 137 or 47 miles. The distance from Utica to
Albany is 263 - 184 or 79 miles.
b. Given: Buffalo, Utica, Albany, and Syracuse are
collinear.
Buffalo is the farthest west.
Albany is 126 miles from Syracuse.
Albany is 263 miles from Buffalo.
Buffalo is 137 miles from Syracuse.
Buffalo is 184 miles from Utica.
Prove: The cities from west to east are Buffalo,
Syracuse, Utica, and Albany.
It is 137 miles from Buffalo to Syracuse.
It is 47 miles from Syracuse to Utica.
It is 79 miles from Utica to Albany.
Proof:
We are given that all of the points are collinear.
Since Syracuse is 137 miles from Buffalo and Albany
is 263 miles from Buffalo, Syracuse is between
Buffalo and Albany. Since Utica is 184 miles from
Buffalo, and Syracuse is 137 miles from Buffalo,
Syracuse is between Utica and Buffalo. Since
Albanyis253milesfromBuffalo,andUticais184
milesfromBuffalo,UticaisbetweenAlbanyand
Buffalo.Therefore,fromeasttowest,thecitiesare
Buffalo,Syracuse,Utica,andAlbany.
Syracuse is 137 miles from Buffalo and Utica is 184
milesfromBuffalo,so,usingtheSegmentAddition
Postulate,Syracuseis184−137,or47milesfrom
Utica. The distance from Buffalo to Albany is 263
miles and the distance from Buffalo to Utica is 184
miles,so,usingtheSegmentAdditionPostulate,the
distance from Utica to Albany is 263 −184, or 79
miles.
PROOF Prove the following.
9.If .
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent segment.
Usethepropertiesthatyouhavelearnedabout
congruent segments and equivalent expressions in
algebra to walk through the proof.
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. SC = HR and HR = AB(Definitionofcongruent
segments)
3. SC = AB (Transitive Property)
4. (Definitionofcongruentsegments)
10.If
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent segments.
Use the properties that you have learned about
congruent segments and equivalent expressions in
algebra to walk through the proof.
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. VZ = VY and WY = XZ (Definition of congruent
segments)
3. VZ = VX + XZ and VY = VW + WY (Seg.ment
Addition Postulate)
4. VX + XZ = VW + WY (Substitution)
5. VX + WY = VW + WY (Substitution)
6. VX = VW (Subtraction Property)
7. VW = VX (Symmetric Property)
8. (Definitionofcongruentsegments.)
11.If Eisthemidpointof and then
.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given a congruent segments
andamidpointofasegment.Usethepropertiesthat
you have learned about congruent segments and
equivalent expressions in algebra to walk through the
proof.
Given: Eisthemidpointof .
Prove:
Proof:
Statements (Reasons)
1. E is the midpoint of . (Given)
2. DE = EF (Definition of midpoint)
3. CD = FG (Definition of congruent segments)
4. CD + DE = EF + FG (Addition Property)
5. CE = CD + DE and EG = EF + FG (Segment
Addition Postulate)
6. CE = EG (Substitution)
7. (Definition of congruent segments)
12.If B is the midpoint of , D is the midpoint of ,
and thenAE = 4AB.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove.Here,youaregivencongruentsegments,two
midpoints of segments. Use the properties that you
have learned about congruent segments and
equivalent expressions in algebra to walk through the
proof.
Given: B is the midpoint of , D is the midpoint of
, and .
Prove: AE = 4AB
Proof:
Statements (Reasons)
1. B is the midpoint of , D is the midpoint of
, and . (Given)
2. AB = BC and CD = DE (Definition of midpoint)
3. AB = DE (Definition of congruent segments)
4. AC = AB + BC and CE = CD + DE (Segment
Addition Postulate)
5. AE = AC + CE (Segment Addition Postulate)
6. AE = AB + BC + CD + DE (Substitution)
7. AE = AB + AB + AB + AB (Substitution)
8. AE = 4AB (Substitution)
13.OPTICAL ILLUSION and
AC + CF + FE = GI + IL + LK.
a. Prove that
b. Justify your proof using measurement. Explain
your method.
SOLUTION:
a. You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent segments,
equal distances for sum of 3 sides. Use the
properties that you have learned about congruent
segments and equivalent expressions in algebra to
walk through the proof.
Given: AC + CF + FE = GI +
IL + LK
Prove:
Proof:
Statements (Reasons)
1. AC + CF + FE = GI + IL +
LK
(Given)
2. AC + CF + FE = AC + IL + LK (Substitution)
3. AC −AC + CF + FE = AC −AC + IL + LK
(Subtraction Property)
4. CF + FE = IL + LK (Substitution Property)
5. CF + FE = IL + FE (Substitution)
6. CF + FE – FE = IL + FE – FE (Subtraction
Property)
7. CF = IL (Substitution Property)
8. (Definitionofcongruentsegments)
b. Sample answer: When using the student edition,
themeasuresof and are1.5incheslong, so
the two segments are congruent.
14.CONSTRUCTION Construct a segment that is
twice as long as .
Explain how the Segment Addition Postulate can be
used to justify your construction.
SOLUTION:
Use a compass and straightedge for the construction.
I placed an initial point A on a line and
constructed a point B on the line so that AB is equal
to PQ.
Using point B as an initial point, I marked point C on
the line so that BC is also equal to PQ. The length of
the whole segment AC is AB + BC according to the
Additional Postulate and AB = BC = PQ.
Using substitution AC = PQ + PQ, or AC = 2PQ, so
is twice as long as .
15.BASEBALL Use the diagram of a baseball
diamond shown..
a. On a baseball field,. P is the
midpointof and . Using a two-column
proof, prove that .
b. The distance from home plate to second base is
127.3 feet. What is the distance from first base to
second base?
SOLUTION:
a.You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent segments
andthemidpointoftwosegments.Usethe
properties that you have learned about congruent
segments and equivalent expressions in algebra to
walk through the proof.
Given: ; P is the midpoint
of .
Prove:
Proof:
Statements (Reasons)
1. ; P is the midpoint of ; P is the
midpointof (Given)
2. SH = TF (Definition of congruent segments)
3. SP = PH, TP = PF (Definition. of Midpoint)
4. SH = SP + PH, TF = TP + PF (Segment Addition
Postulate)
5. SP + PH = TP + PF (Substitution)
6. SP + SP = TP + TP (Substitution)
7. 2SP = 2TP (Substitution)
8. SP = TP (Division Property)
9. (Definitionofcongruentsegments)
b. The home plate, first base and second base form
an isosceles triangle with right angle at F.
Let HF = x, then SF = x.
Use the Pythagorean Theorem.
The distance from first base to second base is about
90 ft.
16.MULTIPLE REPRESENTATIONS A is the
midpoint of B is the midpoint of , and
C is the midpoint of .
a. GEOMETRIC Make a sketch to represent this
situation.
b. ALGEBRAIC Make a conjecture as to the
algebraic relationship between PC and PQ.
c. GEOMETRIC Copy segment from your
sketch. Then construct points B and C on .
Explain how you can use your construction to support
your conjecture.
d. CONCRETE Use a ruler to draw a segment
congruent to from your sketch and to draw
points B and C on . Use your drawing to support
your conjecture.
e. LOGICAL Prove your conjecture.
SOLUTION:
a.
b.WeknowthatPA=AQand,PC=CB,and
PB=BA. Also, by segment addition
PC+CB+BA+AQ=PQ.
ThenPC+CB+BA+AQ=PQ.
PC+PC+PB+PA=PQ(SubstitutePCforCB,PB
for BA, PA for AQ)
PC+PC+(PC+CB)+(PC+CB+BA) = PQ
(SubstitutePC+CBforPB,PC+CB+BAforPA)
PC+PC+PC+PC+PC+PC+(PC+CB) = PQ
(Substitute PC for CB and PC + CB for BA)
PC+PC+PC+PC+PC+PC+PC+PC = PQ
(Substitute PC for CB)
Thus, 8 PC = PQ.
c.
I can measure and mark off segments of that
length along and count how many segments
were formed.
d.
8PC = PQ
e.You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given midpoints of three
segments,Use the properties that you have learned
about congruent segments and equivalent expressions
in algebra to walk through the proof.
Given: A is the midpoint of B is the midpoint of
and C is the midpoint of .
Prove: 8PC = PQ
Statements (Reasons)
1. A is the midpoint of B is the midpoint of
and C is the midpoint of . (Given)
2. PA = AQ, PB = BA, PC = CB (Definition of
Midpoint)
3. PC + CB = PB (Segment Addition Postulate)
4. PC + PC = PB (Substitution)
5. 2PC = PB (Substitution.)
6. PB + BA = PA (Segment Addition Postulate)
7. PB + PB = PA (Substitution.)
8. 2PB = PA (Addition Property)
9. 2 (2PC) = PA (Substitution.)
10. 4 PC = PA (Substitution.)
11. PA + AQ = PQ (Segment Addition Postulate)
12. PA + PA = PQ (Substitution.)
13. 2PA = PQ (Substitution.)
14. 2(4PC) = PQ (Substitution.)
15. 8PC = PQ (Substitution.)
17.CCSS CRITIQUE In the diagram,
.
Examine the conclusions made by Leslie and
Shantice. Is either of them correct?
SOLUTION:
Neither is correct. Since
,then by the
Transitive Property of Congruence. Leslie indicated
the correct property, but applied it incorrectly.
Stantice stated the Transitive Property correct, but
indicated that the property was the Reflexive
Property.
18.CHALLENGE ABCD is a square. Prove that
.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given a square.Use the
properties that you have learned about congruent
segments, squares, and equivalent expressions in
algebra to walk through the proof.
Given: ABCD is a square.
Prove:
Statements (Reasons)
1. ABCD is a square. (Given)
2. AB = BC = CD = DA (Definition of a square)
3.
(Pythagorean Theorem)
4. (Substitution)
5. (TransitiveProperty)
6. (Square Root Property)
7. (By definition, length must be
positive.)
8. AC = BD (Definition of Square Root)
9. (Definition.ofcongruentsegments)
19.WRITING IN MATH Does there exist an Addition
Property of Congruence? Explain.
SOLUTION:
There does not exist an Addition Property of
Congruence. Congruence refers to segments.
Segments cannot be added, only the measures of
segments.
20.REASONING Classify the following statement as
true or false. If false, provide a counterexample.
If A, B, C, D, and E are collinear with B between
A and C, C between B and D, and D between C
and E, and AC = BD = CE, then AB = BC = DE.
SOLUTION:
Thestatementisfalse.
LetAC= BD = CE = 10, then AB = BC = DE = 10.
However, AB = 7, BC = 3, CD = 7 and DE = 3.
21.OPEN ENDED Draw a representation of the
Segment Addition Postulate in which the segment is
two inches long, contains four collinear points, and
contains no congruent segments.
SOLUTION:
Draw a segment two inches long. label points A, B,
C, and D such that the distance between any of the
pointsisdifferent.
22.WRITING IN MATH Compare and contrast
paragraph proofs and two-column proofs.
SOLUTION:
Paragraph proofs and two-column proofs both use
deductive reasoning presented in a logical order along
with the postulates, theorems, and definitions used to
supportthestepsoftheproofs.
Paragraph proofs are written as a paragraph with the
reasons for each step incorporated into the
sentences.
Two-column proofs are numbered and itemized.
Each step of the proof is provided on a separate line
with the support for that step in the column beside
the step.
23.ALGEBRA The chart below shows annual
recycling by material in the United States. About
how many pounds of aluminum are recycled each
year?
A 7.5
B 15,000
C 7,500,000
D 15,000,000,000
SOLUTION:
From the figure we can see that 7.5 million tons of
aluminum is recycled each year.
Convert 7.5 million to pounds.
1 ton = 2000 lb.
7,500,000×2000=15,000,000,000
The correct choice is D.
24.ALGEBRA Which expression is equivalent to
?
F
G
H
J
SOLUTION:
The correct choice is G.
25.SHORT RESPONSE The measures of two
complementary angles are in the ratio 4:1. What is
the measure of the smaller angle?
SOLUTION:
If two angles are complementary then their sum is
90°.
Let x be the measure of the smaller angle. Then the
measure of the larger angle is 4x.
26.SAT/ACT Julie can word process 40 words per
minute. How many minutes will it take Julie to word
process 200 words?
A 0.5
B 2
C 5
D 10
E 12
SOLUTION:
Divide 200 by 40.
200÷40=5
The correct choice is C.
27.PROOF Write a two-column proof.
Given: AC = DF
AB = DE
Prove: BC = EF
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given the measures of two
segments are equal. Use the properties that you have
learned about congruent segments and equivalent
expressions in algebra to walk through the proof.
Given: AC = DF, AB = DE
Prove: BC = EF
Proof:
Statements (Reasons)
1. AC = DF, AB = DE (Given)
2. AC = AB + BC; DF = DE + EF (Segment
Addition Postulate)
3. AB + BC = DE + EF (Substitution)
4. BC = EF (Subtraction Property)
28.MODELS Brian is using six squares of cardboard to
form a rectangular prism. What geometric figure do
the pieces of cardboard represent, and how many
lines will be formed by their intersections?
SOLUTION:
The pieces of cardboard represent planes. There are
12 edges in a rectangular prism. So, 12 lines will be
formed.
29.PATTERN BLOCKS Pattern blocks can be
arranged to fit in a circular pattern without leaving
spaces. Remember that the measurement around a
fullcircleis360°.Determinethedegreemeasureof
the numbered angles shown below.
SOLUTION:
Count the number of parts in to which the circle is
dividedanddivided360°bythenumberofparts.
360°÷6=60°
360°÷12=30°
360°÷4=90°
360°÷6=60°
360°÷3=120°
360°÷6=60°
Simplify.
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
ALGEBRA Find x.
34.
SOLUTION:
Thesumofthetwoanglesis90°.
(8x+1)°+(5x – 2)°=90°
(13x – 1) = 90
13x = 91
Divide both sides by 13.
x = 7
35.
SOLUTION:
The two angles form a linear pair. So, the sum of
theiranglesis180°.
14x + 8x + 4 = 180
22x + 4 = 180
22x = 176
x = 8
36.
SOLUTION:
2x + 4x = 90
6x = 90
x = 15
1.CCSSARGUMENTSCopy and complete the
proof.
Given:
Prove:
Proof:
SOLUTION:
The 1st row is the information given.
The 2nd row the definition of congruent segment,
changes congruent symbols changed to equal signs
and the remove the bars above the segment to
indicate the lengths of the segments.
The 3rd row is found adding the two congruent
segments.
The 4th row is uses the Segment Addition Postulate
to rewrite the segments.
The 5th row is substitution to replace the segments
with equivalent segments.
The 6th row is replacing the = with congruent
symbols and changing the segments lengths to
segment using the definition of congruent segments. .
2.PROOF Prove the following.
Given:
Prove:
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent segments.
You need to find a way to relate the smaller segment
with the larger segments. Use the properties that you
have learned about congruent segments and
equivalent expressions in algebra to walk through the
proof.
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. WX =YZ ( Definition of congruent segments)
3. XY = XY (Reflexive Property)
4. WX + XY = XY + YZ (Addition Property)
5. WY = WX + XY; XZ = XY + YZ (Segment Addition
Postulate)
6. WY = XZ (Substitution.)
7. (Definitionofcongruentsegments)
3.SCISSORS Refer to the diagram shown. is
congruent to . iscongruentto . Prove
that AP + DP = CP + BP.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two pairs of congruent
segments. Use the properties that you have learned
about congruent segments and equivalent expressions
in algebra to walk through the proof.
Given:
Prove: AR + DP = CP + BP
Proof:
Statements (Reasons)
1. (Given)
2. AR = CP, = DP = BP(Definitionofcongruent
segments)
3. AR + DP = CP + DP(AdditionProperty)
4. AR + DP = CP + BP(Substitution.)
4.CCSS ARGUMENTS Copy and complete the
proof.
Given: C is the midpoint of .
C is the midpoint of .
Prove:
Proof:
SOLUTION:
The 1st contains the given information about the
midpoints of segments and segment congruence.
The 2nd row is uses the midpoint of each segment to
write equivalence using the definition of midpoints.
The 3rd row changes the segment congruence to
distance equivalence using the definition of congruent
segments.
The4throwrewritestwoequalsegmentseachwith
two parts.
The 5th row is replaces or substitutes the two equal
segments with their parts.
The 6th row is replaces or substitutes segments with
congruent segments.
The 7th row is simplifying or replacing segments by
combining them.
The 8th row is found by dividing by 2.
The 9th row is changing segment length to segment
congruence using definition of congruent segments.
5.TILING A tile setter cuts a piece of tile to a desired
length. He then uses this tile as a pattern to cut a
second tile congruent to the first. He uses the first
two tiles to cut a third tile whose length is the sum of
the measures of the first two tiles. Prove that the
measure of the third tile is twice the measure of the
first tile.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent tiles and
another tile which is equal in length to the sum of the
lengths of the other tiles. You need to find a way to
relate the large tile with the first small tile. Use the
properties that you have learned about congruent
segments and equivalent expressions in algebra to
walk through the proof.
Given: , AB + CD = EF
Prove: 2AB = EF
Statements (Reasons)
1. , AB + CD = EF (Given)
2. AB = CD (Definition of congruent segment)
3. AB + AB = EF (Substitution)
4. 2AB = EF (Substitution Property)
CCSSARGUMENTSProveeachtheorem.
6.Symmetric Property of Congruence (Theorem 2.2)
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent congruent
segment. You need to find a way to relate the
congruent segments to itself in different order. Use
the properties that you have learned about congruent
segments and equivalent expressions in algebra to
walk through the proof.
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. AB = CD (Definition of congruent segments)
3. CD = AB (Symmetric. Properties)
4. (Definitionofcongruentsegments)
7.Reflexive Property of Congruence (Theorem 2.2)
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are segment. You need to find a
way to relate the the segment to itself. Use the
properties that you have learned about congruent
segments and equivalent expressions in algebra to
walk through the proof.
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. AB = AB (Reflexive. Property)
3. (Definitionofcongruentsegments.)
8.TRAVEL Four cities in New York are connected
by Interstate 90: Buffalo, Utica, Albany, and
Syracuse. Buffalo is the farthest west.
•Albany is 126 miles from Syracuse and 263 miles
from Buffalo.
•Buffalo is 137 miles from Syracuse and 184 miles
from Utica.
a. Draw a diagram to represent the locations of the
cities in relation to each other and the distances
between each city. Assume that Interstate 90 is
straight.
b. Write a paragraph proof to support your
conclusion.
SOLUTION:
a.Use the information given to determine the order
of the cities. Since Buffalo is furthest west, start with
the information given about Buffalo. Albany is 263
miles from Buffalo. Buffalo is 137 miles from
Syracuse. So Syracuse must be between Buffalo and
Albany. Next, Buffalo is 184 miles from Utica. So,
Utica must be between Syracuse and Albany.
Next, find the distances between each consecutive
pair of cities. The distance from Syracuse to Utica is
184 - 137 or 47 miles. The distance from Utica to
Albany is 263 - 184 or 79 miles.
b. Given: Buffalo, Utica, Albany, and Syracuse are
collinear.
Buffalo is the farthest west.
Albany is 126 miles from Syracuse.
Albany is 263 miles from Buffalo.
Buffalo is 137 miles from Syracuse.
Buffalo is 184 miles from Utica.
Prove: The cities from west to east are Buffalo,
Syracuse, Utica, and Albany.
It is 137 miles from Buffalo to Syracuse.
It is 47 miles from Syracuse to Utica.
It is 79 miles from Utica to Albany.
Proof:
We are given that all of the points are collinear.
Since Syracuse is 137 miles from Buffalo and Albany
is 263 miles from Buffalo, Syracuse is between
Buffalo and Albany. Since Utica is 184 miles from
Buffalo, and Syracuse is 137 miles from Buffalo,
Syracuse is between Utica and Buffalo. Since
Albanyis253milesfromBuffalo,andUticais184
milesfromBuffalo,UticaisbetweenAlbanyand
Buffalo.Therefore,fromeasttowest,thecitiesare
Buffalo,Syracuse,Utica,andAlbany.
Syracuse is 137 miles from Buffalo and Utica is 184
milesfromBuffalo,so,usingtheSegmentAddition
Postulate,Syracuseis184−137,or47milesfrom
Utica. The distance from Buffalo to Albany is 263
miles and the distance from Buffalo to Utica is 184
miles,so,usingtheSegmentAdditionPostulate,the
distance from Utica to Albany is 263 −184, or 79
miles.
PROOF Prove the following.
9.If .
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent segment.
Usethepropertiesthatyouhavelearnedabout
congruent segments and equivalent expressions in
algebra to walk through the proof.
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. SC = HR and HR = AB(Definitionofcongruent
segments)
3. SC = AB (Transitive Property)
4. (Definitionofcongruentsegments)
10.If
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent segments.
Use the properties that you have learned about
congruent segments and equivalent expressions in
algebra to walk through the proof.
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. VZ = VY and WY = XZ (Definition of congruent
segments)
3. VZ = VX + XZ and VY = VW + WY (Seg.ment
Addition Postulate)
4. VX + XZ = VW + WY (Substitution)
5. VX + WY = VW + WY (Substitution)
6. VX = VW (Subtraction Property)
7. VW = VX (Symmetric Property)
8. (Definitionofcongruentsegments.)
11.If Eisthemidpointof and then
.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given a congruent segments
andamidpointofasegment.Usethepropertiesthat
you have learned about congruent segments and
equivalent expressions in algebra to walk through the
proof.
Given: Eisthemidpointof .
Prove:
Proof:
Statements (Reasons)
1. E is the midpoint of . (Given)
2. DE = EF (Definition of midpoint)
3. CD = FG (Definition of congruent segments)
4. CD + DE = EF + FG (Addition Property)
5. CE = CD + DE and EG = EF + FG (Segment
Addition Postulate)
6. CE = EG (Substitution)
7. (Definition of congruent segments)
12.If B is the midpoint of , D is the midpoint of ,
and thenAE = 4AB.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove.Here,youaregivencongruentsegments,two
midpoints of segments. Use the properties that you
have learned about congruent segments and
equivalent expressions in algebra to walk through the
proof.
Given: B is the midpoint of , D is the midpoint of
, and .
Prove: AE = 4AB
Proof:
Statements (Reasons)
1. B is the midpoint of , D is the midpoint of
, and . (Given)
2. AB = BC and CD = DE (Definition of midpoint)
3. AB = DE (Definition of congruent segments)
4. AC = AB + BC and CE = CD + DE (Segment
Addition Postulate)
5. AE = AC + CE (Segment Addition Postulate)
6. AE = AB + BC + CD + DE (Substitution)
7. AE = AB + AB + AB + AB (Substitution)
8. AE = 4AB (Substitution)
13.OPTICAL ILLUSION and
AC + CF + FE = GI + IL + LK.
a. Prove that
b. Justify your proof using measurement. Explain
your method.
SOLUTION:
a. You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent segments,
equal distances for sum of 3 sides. Use the
properties that you have learned about congruent
segments and equivalent expressions in algebra to
walk through the proof.
Given: AC + CF + FE = GI +
IL + LK
Prove:
Proof:
Statements (Reasons)
1. AC + CF + FE = GI + IL +
LK
(Given)
2. AC + CF + FE = AC + IL + LK (Substitution)
3. AC −AC + CF + FE = AC −AC + IL + LK
(Subtraction Property)
4. CF + FE = IL + LK (Substitution Property)
5. CF + FE = IL + FE (Substitution)
6. CF + FE – FE = IL + FE – FE (Subtraction
Property)
7. CF = IL (Substitution Property)
8. (Definitionofcongruentsegments)
b. Sample answer: When using the student edition,
themeasuresof and are1.5incheslong, so
the two segments are congruent.
14.CONSTRUCTION Construct a segment that is
twice as long as .
Explain how the Segment Addition Postulate can be
used to justify your construction.
SOLUTION:
Use a compass and straightedge for the construction.
I placed an initial point A on a line and
constructed a point B on the line so that AB is equal
to PQ.
Using point B as an initial point, I marked point C on
the line so that BC is also equal to PQ. The length of
the whole segment AC is AB + BC according to the
Additional Postulate and AB = BC = PQ.
Using substitution AC = PQ + PQ, or AC = 2PQ, so
is twice as long as .
15.BASEBALL Use the diagram of a baseball
diamond shown..
a. On a baseball field,. P is the
midpointof and . Using a two-column
proof, prove that .
b. The distance from home plate to second base is
127.3 feet. What is the distance from first base to
second base?
SOLUTION:
a.You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two congruent segments
andthemidpointoftwosegments.Usethe
properties that you have learned about congruent
segments and equivalent expressions in algebra to
walk through the proof.
Given: ; P is the midpoint
of .
Prove:
Proof:
Statements (Reasons)
1. ; P is the midpoint of ; P is the
midpointof (Given)
2. SH = TF (Definition of congruent segments)
3. SP = PH, TP = PF (Definition. of Midpoint)
4. SH = SP + PH, TF = TP + PF (Segment Addition
Postulate)
5. SP + PH = TP + PF (Substitution)
6. SP + SP = TP + TP (Substitution)
7. 2SP = 2TP (Substitution)
8. SP = TP (Division Property)
9. (Definitionofcongruentsegments)
b. The home plate, first base and second base form
an isosceles triangle with right angle at F.
Let HF = x, then SF = x.
Use the Pythagorean Theorem.
The distance from first base to second base is about
90 ft.
16.MULTIPLE REPRESENTATIONS A is the
midpoint of B is the midpoint of , and
C is the midpoint of .
a. GEOMETRIC Make a sketch to represent this
situation.
b. ALGEBRAIC Make a conjecture as to the
algebraic relationship between PC and PQ.
c. GEOMETRIC Copy segment from your
sketch. Then construct points B and C on .
Explain how you can use your construction to support
your conjecture.
d. CONCRETE Use a ruler to draw a segment
congruent to from your sketch and to draw
points B and C on . Use your drawing to support
your conjecture.
e. LOGICAL Prove your conjecture.
SOLUTION:
a.
b.WeknowthatPA=AQand,PC=CB,and
PB=BA. Also, by segment addition
PC+CB+BA+AQ=PQ.
ThenPC+CB+BA+AQ=PQ.
PC+PC+PB+PA=PQ(SubstitutePCforCB,PB
for BA, PA for AQ)
PC+PC+(PC+CB)+(PC+CB+BA) = PQ
(SubstitutePC+CBforPB,PC+CB+BAforPA)
PC+PC+PC+PC+PC+PC+(PC+CB) = PQ
(Substitute PC for CB and PC + CB for BA)
PC+PC+PC+PC+PC+PC+PC+PC = PQ
(Substitute PC for CB)
Thus, 8 PC = PQ.
c.
I can measure and mark off segments of that
length along and count how many segments
were formed.
d.
8PC = PQ
e.You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given midpoints of three
segments,Use the properties that you have learned
about congruent segments and equivalent expressions
in algebra to walk through the proof.
Given: A is the midpoint of B is the midpoint of
and C is the midpoint of .
Prove: 8PC = PQ
Statements (Reasons)
1. A is the midpoint of B is the midpoint of
and C is the midpoint of . (Given)
2. PA = AQ, PB = BA, PC = CB (Definition of
Midpoint)
3. PC + CB = PB (Segment Addition Postulate)
4. PC + PC = PB (Substitution)
5. 2PC = PB (Substitution.)
6. PB + BA = PA (Segment Addition Postulate)
7. PB + PB = PA (Substitution.)
8. 2PB = PA (Addition Property)
9. 2 (2PC) = PA (Substitution.)
10. 4 PC = PA (Substitution.)
11. PA + AQ = PQ (Segment Addition Postulate)
12. PA + PA = PQ (Substitution.)
13. 2PA = PQ (Substitution.)
14. 2(4PC) = PQ (Substitution.)
15. 8PC = PQ (Substitution.)
17.CCSS CRITIQUE In the diagram,
.
Examine the conclusions made by Leslie and
Shantice. Is either of them correct?
SOLUTION:
Neither is correct. Since
,then by the
Transitive Property of Congruence. Leslie indicated
the correct property, but applied it incorrectly.
Stantice stated the Transitive Property correct, but
indicated that the property was the Reflexive
Property.
18.CHALLENGE ABCD is a square. Prove that
.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given a square.Use the
properties that you have learned about congruent
segments, squares, and equivalent expressions in
algebra to walk through the proof.
Given: ABCD is a square.
Prove:
Statements (Reasons)
1. ABCD is a square. (Given)
2. AB = BC = CD = DA (Definition of a square)
3.
(Pythagorean Theorem)
4. (Substitution)
5. (TransitiveProperty)
6. (Square Root Property)
7. (By definition, length must be
positive.)
8. AC = BD (Definition of Square Root)
9. (Definition.ofcongruentsegments)
19.WRITING IN MATH Does there exist an Addition
Property of Congruence? Explain.
SOLUTION:
There does not exist an Addition Property of
Congruence. Congruence refers to segments.
Segments cannot be added, only the measures of
segments.
20.REASONING Classify the following statement as
true or false. If false, provide a counterexample.
If A, B, C, D, and E are collinear with B between
A and C, C between B and D, and D between C
and E, and AC = BD = CE, then AB = BC = DE.
SOLUTION:
Thestatementisfalse.
LetAC= BD = CE = 10, then AB = BC = DE = 10.
However, AB = 7, BC = 3, CD = 7 and DE = 3.
21.OPEN ENDED Draw a representation of the
Segment Addition Postulate in which the segment is
two inches long, contains four collinear points, and
contains no congruent segments.
SOLUTION:
Draw a segment two inches long. label points A, B,
C, and D such that the distance between any of the
pointsisdifferent.
22.WRITING IN MATH Compare and contrast
paragraph proofs and two-column proofs.
SOLUTION:
Paragraph proofs and two-column proofs both use
deductive reasoning presented in a logical order along
with the postulates, theorems, and definitions used to
supportthestepsoftheproofs.
Paragraph proofs are written as a paragraph with the
reasons for each step incorporated into the
sentences.
Two-column proofs are numbered and itemized.
Each step of the proof is provided on a separate line
with the support for that step in the column beside
the step.
23.ALGEBRA The chart below shows annual
recycling by material in the United States. About
how many pounds of aluminum are recycled each
year?
A 7.5
B 15,000
C 7,500,000
D 15,000,000,000
SOLUTION:
From the figure we can see that 7.5 million tons of
aluminum is recycled each year.
Convert 7.5 million to pounds.
1 ton = 2000 lb.
7,500,000×2000=15,000,000,000
The correct choice is D.
24.ALGEBRA Which expression is equivalent to
?
F
G
H
J
SOLUTION:
The correct choice is G.
25.SHORT RESPONSE The measures of two
complementary angles are in the ratio 4:1. What is
the measure of the smaller angle?
SOLUTION:
If two angles are complementary then their sum is
90°.
Let x be the measure of the smaller angle. Then the
measure of the larger angle is 4x.
26.SAT/ACT Julie can word process 40 words per
minute. How many minutes will it take Julie to word
process 200 words?
A 0.5
B 2
C 5
D 10
E 12
SOLUTION:
Divide 200 by 40.
200÷40=5
The correct choice is C.
27.PROOF Write a two-column proof.
Given: AC = DF
AB = DE
Prove: BC = EF
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given the measures of two
segments are equal. Use the properties that you have
learned about congruent segments and equivalent
expressions in algebra to walk through the proof.
Given: AC = DF, AB = DE
Prove: BC = EF
Proof:
Statements (Reasons)
1. AC = DF, AB = DE (Given)
2. AC = AB + BC; DF = DE + EF (Segment
Addition Postulate)
3. AB + BC = DE + EF (Substitution)
4. BC = EF (Subtraction Property)
28.MODELS Brian is using six squares of cardboard to
form a rectangular prism. What geometric figure do
the pieces of cardboard represent, and how many
lines will be formed by their intersections?
SOLUTION:
The pieces of cardboard represent planes. There are
12 edges in a rectangular prism. So, 12 lines will be
formed.
29.PATTERN BLOCKS Pattern blocks can be
arranged to fit in a circular pattern without leaving
spaces. Remember that the measurement around a
fullcircleis360°.Determinethedegreemeasureof
the numbered angles shown below.
SOLUTION:
Count the number of parts in to which the circle is
dividedanddivided360°bythenumberofparts.
360°÷6=60°
360°÷12=30°
360°÷4=90°
360°÷6=60°
360°÷3=120°
360°÷6=60°
Simplify.
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
ALGEBRA Find x.
34.
SOLUTION:
Thesumofthetwoanglesis90°.
(8x+1)°+(5x – 2)°=90°
(13x – 1) = 90
13x = 91
Divide both sides by 13.
x = 7
35.
SOLUTION:
The two angles form a linear pair. So, the sum of
theiranglesis180°.
14x + 8x + 4 = 180
22x + 4 = 180
22x = 176
x = 8
36.
SOLUTION:
2x + 4x = 90
6x = 90
x = 15
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2-7 Proving Segment Relationships
