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You will explore theorems and properties about parallelograms, special quadrilaterals that have parallel sides, and rectangles, special parallelograms with ...
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In this unit, you will take an in-depth look at quadrilaterals, parallelograms, and rectangles. You will explore theorems and properties about parallelograms, special quadrilaterals that have parallel sides, and rectangles, special parallelograms with right angles.
Quadrilaterals Parallelograms
More Properties of Parallelograms
Summary of Properties of Parallelograms Rectangles
quadrilateral – A quadrilateral is a polygon with four sides.
Several examples of quadrilaterals are shown below.
Here are some figures that are NOT quadrilaterals.
In figure A , the four-sided figure is not a polygon (not a closed figure).
In figure B , look closely to find four sides; but two sides overlap eliminating this figure as a single polygon.
In figure C , there are only two segments, and the other side is one curved line.
In figured D , there are no distinct line segments.
Given: (This is a short way to write that quadrilateral is a parallelogram.
Prove: ;
Statements Reasons
es of parallel lines are congruent. (Rotate the parallelogram 90-degrees clockwise to visualize this better.)
*Note: The theorem numbers are not written in the reasons. You can choose between writing out the entire theorem, an abbreviated version of the theorem, or just the theorem number.
The opposite angles of a parallelogram are Theorem 16-B congruent.
The opposite sides of a parallelogram are Theorem 16-A congruent.
Example 1 : How does Theorem 10-L support Theorem 16-C? Theorem 10-L states that consecutive interior angles of parallel lines are supplementary.
Angles K and L are consecutive angles for parallel lines LM and KN , thus they are supplementary. Angles N and M are consecutive angles for parallel lines LM and KN , thus they are supplementary.
Angles L and M are consecutive angles for parallel lines KL and NM , thus they are supplementary.
Angles K and N are consecutive angles for parallel lines KL and NM , thus they are supplementary.
The consecutive pairs of angles of a Theorem 16-C parallelogram are supplementary.
Example 2 : Given LMPN. Solve for s , t , v , w , and x. Also determine the measure of angle LMN.
To find s , theorem 16-A states that the opposite sides of a parallelogram are congruent.
47 3 7 54 3 18 1
s KL NM s s s
To find t , recall that the alternate interior angles of parallel lines are congruent.
7 3 52 7 4 7
t m MLN m LNK t t
To find v , theorem 16-D states that the diagonals of a parallelogram bisect each other.
v QM * KQ QM v
Either diagonal of a parallelogram separates the parallelogram into two congruent triangles.
Theorem 16-E
63 °^ (7^ t^ +^ 3)° 108 °
52 °
3 s − 7 x ° w °^ v
To find w , first recall that vertical angles are congruent.
108 108
m LQM * m LQM m KQN m KQN
Then, recall the Triangle Sum Theorem; that is, the sum of the angles in a triangle equals 180.
108 52 180 180 160 180 20
w * m KQN m QNK m NKQ w w
To find x , recall the Exterior Angle Sum Theorem; that is, the exterior angle of a triangle equals the sum of the two remote interior angles.
108 63 45 5
x m LQM x m KLQ x x
To find m ∠ LMN ,first determine the m ∠ LKN by recalling the Angle Addition Postulate, and then apply theorem 16-B; that is, opposite angles of a parallelogram are congruent.
45 20 (Angle Addition Postulate) 65 65 (Theorem 16-B
m LKN m LKN x w m LKN m LMN m LKN m LMN
We can plan an approach for the proof of this theorem by looking at the end results and working backwards.
We want to end with a parallelogram.
We need to show that the opposite sides are parallel.
If we add a diagonal, an auxiliary line, we can use the theorems associated with parallel lines and transversals
In particular, we need to recall theorem 11-A: “If the alternate interior angles of two lines cut by a transversal are congruent, then the lines are parallel.”
So, let’s get started!
In a quadrilateral if both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.
Theorem 16-F
Given: Quadrilateral QRST RS QT RQ ST
Prove: Quadrilateral QRST is a parallelogram.
Statements Reasons
two lines cut by a transversal are congruent, then the lines are parallel. (Tm 11-A)
kwi
se.)
*Tm is the abbreviation for “Theorem”.
In a quadrilateral if both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram.
Theorem 16-G
Example 5 : Apply Theorem 16-I to determine if quadrilateral FGHK , with vertices F(–1,7), G(3,5), H(2,0), and K(–2, 2), is a parallelogram.
First we’ll check a pair of opposites sides to see if they are parallel; then, we’ll check to see if they are equal in measures.
of 2 0 2 1 2 2 4 2 of 7 5 2 1 1 3 4 2
m KH
m FG
The slopes are the same; therefore, KH FG.
In a quadrilateral if one pair of opposite sides is both congruent and parallel, then the quadrilateral is a parallelogram.
Theorem 16-I
x
y
2 2
The segments are the same length, therefore they are congruent.
We have proven that KH FG and KH ≅ FG ; therefore, by Theorem 16-I, quadrilateral FGHK is a parallelogram.
In a parallelogram, what kind of triangles is created by either of the diagonals? A diagonal divides a parallelogram into two congruent triangles. ( Theorem 16-F )
Is a quadrilateral a parallelogram? Yes, if both pairs of opposite sides are congruent. ( Theorem 16-G )
Is a quadrilateral a parallelogram? Yes, if its diagonals bisect each other. ( Theorem 16-H )
Is a quadrilateral a parallelogram? Yes, if one pair of sides are both parallel and congruent. ( Theorem 16-I )
m R m S m S m T m Q m T m R m Q
Theorem 16 - C
Theorem 16 -D
Theorem 16 -E
rectangle – A rectangle is a parallelogram with four right angles.
We will construct a rectangle that is 4 inches long and 3 inches wide. We will then check to see if the diagonals are congruent.
Step 1 : Use a straightedge to draw line b. Label a point H on line b. On a ruler lay the metal point of the compass at 0 inches and then open your compass so that the pencil point touches the 4-inch mark on the ruler. Place the metal point of the compass at point H and mark point G so that GH measures 4 inches.
Step 2 : Construct perpendicular lines to b at points G and H. Label the lines p and s.
Step 3 : Using a ruler for reference, open the compass so that the distance between the metal point and the pencil point is 3 inches. Place the metal point of the compass at point G and mark off 3 inches on line p. Then move the metal point of the compass to point H and mark off 3 inches on line s. Draw AB.
If a parallelogram is a rectangle, then its diagonals are congruent. Theorem 16-J
p (^) s
G H^ b
G H^ b
4 in
p (^) s
G H^ b
3 in
