ParallelogramsParallelogramsParallelogramsParallelograms

5-15-1

EXAMPLE 1 Use properties of parallelograms

ALGEBRA Find the values of x and y.

ABCD is a parallelogram by the definition of a parallelogram. Use Theorem 8.3 to find the value of x.

Opposite sides of a are .AB = CD

Substitute x + 4 for AB and 12 for CD.x + 4 = 12Subtract 4 from each side.x = 8

By Theorem 8.4, A C, or m A = m C. So, y ° = 65°.

In ABCD, x = 8 and y = 65.ANSWER

GUIDED PRACTICE for Example 1

Find FG and m G.1.

SOLUTION

Opposite sides of a are .FG = HE

x = 8

By Theorem 8.4, E G, or m E = m G. So, G ° = 60°.

In FEHG, FG = 8 and m G = 60°.ANSWER

GUIDED PRACTICE for Example 1

Find the values of x and y.2.

SOLUTION

Opposite sides of a are .JK = ML

Substitute 18 for JK and y + 3 for ML.18 = y + 3

Subtract 3 from each side.15 = y

By Theorem 8.4, J L, or m J = m L.

Substitute2x = 50

Divide 2 from each side.x = 25

In JKLM, x = 25 and y = 15.ANSWER

EXAMPLE 2 Use properties of parallelograms

So, m ADC + m BCD = 180°. Because m ADC = 110°, m BCD =180° –110° = 70°.

SOLUTION

By Theorem 8.5, the consecutive angle pairs in ABCD are supplementary.

Desk Lamp

As shown, part of the extending arm of a desk lamp is a parallelogram. The angles of the parallelogram change as the lamp is raised and lowered. Find m BCD when m ADC = 110°.

EXAMPLE 3 Standardized Test Practice

SOLUTION

By Theorem 8 .6, the diagonals of a parallelogram bisect each other. So, P is the midpoint of diagonals LN and OM . Use the Midpoint Formula.

The correct answer is A.ANSWER

= Coordinates of midpoint P of OM 4 + 02

7 + 02

,( ) = 72

,2 ( )

GUIDED PRACTICE for Examples 2 and 3

NM3.

SOLUTION

By Theorem 8 .6, the diagonals of a parallelogram bisect each other. So, N is the midpoint of diagonals KM .

KN = NM

Substitute2 = NM

Find the indicated measure in JKLM.

GUIDED PRACTICE for Examples 2 and 3

Find the indicated measure in JKLM.

KM 4.

SOLUTION

By theorem 8.6KM = KN + NM

SubstituteKM = 2 + 2

AddKM = 4

GUIDED PRACTICE for Examples 2 and 3

m JML 5.

SOLUTION

By Theorem 8.5, the consecutive angle pairs in JKLM are supplementary.

So, m KJM + m JML = 180°.

Because m KJM = 110°,m JML =180° –110° = 70°.

Find the indicated measure in JKLM.

GUIDED PRACTICE for Examples 2 and 3

Find the indicated measure in JKLM.

m KML 6.

SOLUTION

m JML = m KMJ + m KNL

Substitute70° = 30° + m KML

Subtract40° = m KML

EXAMPLE 2 Identify a parallelogram

ARCHITECTURE

The doorway shown is part of a building in England. Over time, the building has leaned sideways. Explain how you know that SV = TU.

SOLUTION

In the photograph, ST UV and ST UV. By Theorem 8.9, quadrilateral STUV is a parallelogram. By Theorem 8.3, you know that opposite sides of a parallelogram are congruent. So, SV = TU.

EXAMPLE 3 Use algebra with parallelograms

ALGEBRA For what value of x is quadrilateral CDEF a parallelogram?

SOLUTION

By Theorem 8.10, if the diagonals of CDEF bisect each other, then it is a parallelogram. You are given that CN EN . Find x so that FN DN .

EXAMPLE 3 Use algebra with parallelograms

Set the segment lengths equal.FN = DN

Substitute 5x –8 for FN and 3x for DN.5x – 8 = 3x

Subtract 3x from each side.2x – 8 = 0

Add 8 to each side.2x = 8

Divide each side by 2.x = 4

When x = 4, FN = 5(4) –8 = 12 and DN = 3(4) = 12.

Quadrilateral CDEF is a parallelogram when x = 4.

ANSWER

GUIDED PRACTICE for Examples 2 and 3

What theorem can you use to show that the quadrilateral is a parallelogram?

2.

In the graphic, two opposite sides are equal, i.e, 30m each and parallel, Therefore, the quadrilateral is a parallelogram. By theorem 8.9.

ANSWER

GUIDED PRACTICE for Examples 2 and 3

What theorem can you use to show that the quadrilateral is a parallelogram?

3.

Two pairs of opposite sides are equal.

Therefore, the quadrilateral is a parallelogram. By theorem 8.7

ANSWER

GUIDED PRACTICE for Examples 2 and 3

What theorem can you use to show that the quadrilateral is a parallelogram?

4.

By theorem 8.8, if the opposite angles are Congruent, the quadrilateral is a parallelogram.

ANSWER

GUIDED PRACTICE for Examples 2 and 3

For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning.

5.

SOLUTION

[ Diagonals in bisect each other ] By Theorem 8.62x = 10 – 3x

Add 3x to each side5x = 10

Divide each side by 5x = 2

EXAMPLE 2 Identify a parallelogram

ARCHITECTURE

The doorway shown is part of a building in England. Over time, the building has leaned sideways. Explain how you know that SV = TU.

SOLUTION

In the photograph, ST UV and ST UV. By Theorem 8.9, quadrilateral STUV is a parallelogram. By Theorem 8.3, you know that opposite sides of a parallelogram are congruent. So, SV = TU.

EXAMPLE 3 Use algebra with parallelograms

ALGEBRA For what value of x is quadrilateral CDEF a parallelogram?

SOLUTION

By Theorem 8.10, if the diagonals of CDEF bisect each other, then it is a parallelogram. You are given that CN EN . Find x so that FN DN .

EXAMPLE 3 Use algebra with parallelograms

Set the segment lengths equal.FN = DN

Substitute 5x –8 for FN and 3x for DN.5x – 8 = 3x

Subtract 3x from each side.2x – 8 = 0

Add 8 to each side.2x = 8

Divide each side by 2.x = 4

When x = 4, FN = 5(4) –8 = 12 and DN = 3(4) = 12.

Quadrilateral CDEF is a parallelogram when x = 4.

ANSWER

GUIDED PRACTICE for Examples 2 and 3

What theorem can you use to show that the quadrilateral is a parallelogram?

2.

In the graphic, two opposite sides are equal, i.e, 30m each and parallel, Therefore, the quadrilateral is a parallelogram. By theorem 8.9.

ANSWER

GUIDED PRACTICE for Examples 2 and 3

What theorem can you use to show that the quadrilateral is a parallelogram?

3.

Two pairs of opposite sides are equal.

Therefore, the quadrilateral is a parallelogram. By theorem 8.7

ANSWER

GUIDED PRACTICE for Examples 2 and 3

What theorem can you use to show that the quadrilateral is a parallelogram?

4.

By theorem 8.8, if the opposite angles are Congruent, the quadrilateral is a parallelogram.

ANSWER

GUIDED PRACTICE for Examples 2 and 3

For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning.

5.

SOLUTION

[ Diagonals in bisect each other ] By Theorem 8.62x = 10 – 3x

Add 3x to each side5x = 10

Divide each side by 5x = 2