8.2
Use Properties of Parallelograms p Find angle and side measures in parallelograms.
Goal
Your Notes
VOCABULARY Parallelogram
THEOREM 8.3 If a quadrilateral is a parallelogram, then its opposite sides are congruent. If PQRS is a parallelogram, then } } > RS and QR > .
R
P
S
THEOREM 8.4 If a quadrilateral is a parallelogram, then its opposite angles are congruent. If PQRS is a parallelogram, then ∠P > and > ∠S.
Example 1
R
P
S
Use properties of parallelograms
Find the values of x and y.
F
x16
Solution FGHJ is a parallelogram by the definition of a parallelogram. Use Theorem 8.3 to find the value of x. FG 5 x165 x5
210
Lesson 8.2 • Geometry Notetaking Guide
y8 J
13
H
for
.
Opposite sides of a ~ are >. Substitute x 1 6 for FG and Subtract 6 from each side.
By Theorem 8.4, ∠F > y8 5 . In ~FGHJ, x 5
G
688
, or m∠F 5
and y 5
. So,
.
Copyright © Holt McDougal. All rights reserved.
8.2
Use Properties of Parallelograms p Find angle and side measures in parallelograms.
Goal
Your Notes
VOCABULARY Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
THEOREM 8.3 If a quadrilateral is a parallelogram, then its opposite sides are congruent. If PQRS is a parallelogram, then } } } } PQ > RS and QR > PS .
R
P
S
THEOREM 8.4 If a quadrilateral is a parallelogram, then its opposite angles are congruent. If PQRS is a parallelogram, then ∠P > ∠R and ∠Q > ∠S.
Example 1
R
P
Use properties of parallelograms
Find the values of x and y.
F
Solution FGHJ is a parallelogram by the definition of a parallelogram. Use Theorem 8.3 to find the value of x. FG 5 HJ x 1 6 5 13 x5 7
S
x16
G
688
y8 J
13
H
Opposite sides of a ~ are >. Substitute x 1 6 for FG and 13 for HJ . Subtract 6 from each side.
By Theorem 8.4, ∠F > ∠H , or m∠F 5 m∠H . So, y8 5 688 . In ~FGHJ, x 5 7 and y 5 68 . 210
Lesson 8.2 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
THEOREM 8.5 If a quadrilateral is a parallelogram, then its consecutive angles are .
x8
y8
y8
x8
P
S
If PQRS is a parallelogram, then x8 1 y8 5
Example 2
R
.
Use properties of a parallelogram
Gates As shown, a gate contains several parallelograms. Find m∠ADC when m∠DAB 5 658.
A D
B
Solution C By Theorem 8.5, the consecutive . So, angle pairs in ~ABCD are m∠ADC 1 m∠DAB 5 . Because m∠DAB 5 658, m∠ADC 5 2 5 . 37
L
Checkpoint Find the indicated
measure in ~KLMN shown at the right. 1. x
M
1238 z8 K
y8 2x 2 3
N
2. y
3. z
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Lesson 8.2 • Geometry Notetaking Guide
211
Your Notes
THEOREM 8.5 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary .
x8
y8
y8
R
x8
P
S
If PQRS is a parallelogram, then x8 1 y8 5 1808 .
Example 2
Use properties of a parallelogram
Gates As shown, a gate contains several parallelograms. Find m∠ADC when m∠DAB 5 658.
A D
B
Solution C By Theorem 8.5, the consecutive angle pairs in ~ABCD are supplementary . So, m∠ADC 1 m∠DAB 5 1808 . Because m∠DAB 5 658, m∠ADC 5 1808 2 658 5 1158 . 37
L
Checkpoint Find the indicated
measure in ~KLMN shown at the right. 1. x x 5 20
M
1238 z8 K
y8 2x 2 3
N
2. y y 5 123
3. z z 5 57
Copyright © Holt McDougal. All rights reserved.
Lesson 8.2 • Geometry Notetaking Guide
211
Your Notes
THEOREM 8.6 R
If a quadrilateral is a parallelogram, then its diagonals each other.
M P
S
}
QM > } PM > Example 3
Use properties of a parallelogram
The diagonals of ~STUV intersect at point W. Find the coordinates of W. In Example 3, you can use either diagonal to find the coordinates of W. SU simplifies Using } calculations because one endpoint is (0, 0).
and
y
T
U W
Solution 1 By Theorem 8.6, the diagonals of a S 1 V each other. parallelogram } } So, W is the of the diagonals TV and SU. Use the . Coordinates of midpoint W of } SU 5 51
1
2
x
2
Checkpoint Complete the following exercises.
4. The diagonals of ~VWXY intersect at point Z. Find the coordinates of Z.
y W
X
Z 1
V
Homework
5. Given that ~FGHJ is a parallelogram, find MH and FH.
F
G 5 M J
212
Lesson 8.2 • Geometry Notetaking Guide
x
Y
1
H
Copyright © Holt McDougal. All rights reserved.
Your Notes
THEOREM 8.6 R
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
M P
S
} QM > SM and } } PM > RM }
Use properties of a parallelogram
Example 3
The diagonals of ~STUV intersect at point W. Find the coordinates of W. In Example 3, you can use either diagonal to find the coordinates of W. SU simplifies Using } calculations because one endpoint is (0, 0).
y
T
U W
Solution 1 By Theorem 8.6, the diagonals of a S 1 V parallelogram bisect each other. } } So, W is the midpoint of the diagonals TV and SU. Use the Midpoint Formula . Coordinates of midpoint W of 610 510 } 5 SU 5 }, } 5 1 3, }
1
2
2
2
2
x
2
Checkpoint Complete the following exercises.
4. The diagonals of ~VWXY intersect at point Z. Find the coordinates of Z.
y W
Z
7 Z 1} , 32
1
2
V
Homework
5. Given that ~FGHJ is a parallelogram, find MH and FH. MH 5 5, FH 5 10
212
Lesson 8.2 • Geometry Notetaking Guide
X
x
Y
1
F
G 5 M J
H
Copyright © Holt McDougal. All rights reserved.
8.3
Show that a Quadrilateral is a Parallelogram Goal
Your Notes
p Use properties to identify parallelograms.
THEOREM 8.7 B
C
of a If both pairs of opposite quadrilateral are congruent, then the A quadrilateral is a parallelogram. } } If AB > and BC > , then ABCD is a parallelogram. THEOREM 8.8
B
of a If both pairs of opposite quadrilateral are congruent, then the quadrilateral is a parallelogram. If ∠A > and ∠B > parallelogram.
Example 1
D
C
A
D
, then ABCD is a
Solve a real-world problem
Basketball In the diagram at the right, } } AB and DC represent adjustable supports of a basketball hoop. } } Explain why AD is always parallel to BC.
A 18 in. 36 in. B
D
Solution The shape of quadrilateral ABCD changes as the adjustable supports move, but its do not change. Both pairs of opposite are congruent, so ABCD is a parallelogram by . } By the definition of a parallelogram, AD i
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36 in.
18 in. C
.
Lesson 8.3 • Geometry Notetaking Guide
213
