Drill Solve for x.

1. x2 + 38 = 3x2 – 12

2. 137 + x = 180

3.

4. Find FE.

5 or –5

43

156

GT Geometry 2/11/13

Use properties of kites to solve problems.

Use properties of trapezoids to solve problems.

Objectives

kite

trapezoid

base of a trapezoid

leg of a trapezoid

base angle of a trapezoid

isosceles trapezoid

midsegment of a trapezoid

Vocabulary

A kite is a quadrilateral with exactly two pairs of congruent consecutive sides.

Example 1: Problem-Solving Application

Lucy is framing a kite with wooden dowels. She uses two dowels that measure 18 cm, one dowel that measures 30 cm, and two dowels that measure 27 cm. To complete the kite, she needs a dowel to place along . She has a dowel that is 36 cm long. About how much wood will she have left after cutting the last dowel?

Solve3

N bisects JM.

Pythagorean Thm.

Pythagorean Thm.

Example 1 Continued

Lucy needs to cut the dowel to be 32.4 cm long. The amount of wood that will remain after the cut is,

36 – 32.4 3.6 cm

Lucy will have 3.6 cm of wood left over after the cut.

Example 1 Continued

Look Back4

Example 1 Continued

To estimate the length of the diagonal, change the side length into decimals and round. , and

. The length of the diagonal is approximately 10 + 22 = 32. So the wood remaining is approximately 36 – 32 = 4. So 3.6 is a reasonable answer.

Check It Out! Example 1

What if...? Daryl is going to make a kite by doubling all the measures in the kite. What is the total amount of binding needed to cover the edges of his kite? How many packages of binding must Daryl buy?

Solve3

Pyth. Thm.

Pyth. Thm.

Check It Out! Example 1 Continued

perimeter of PQRS =

Daryl needs approximately 191.3 inches of binding.One package of binding contains 2 yards, or 72 inches.

In order to have enough, Daryl must buy 3 packages of binding.

Check It Out! Example 1 Continued

packages of binding

Look Back4

Check It Out! Example 1 Continued

To estimate the perimeter, change the side lengths into decimals and round.

, and . The perimeter of the kite is approximately

2(54) + 2 (41) = 190. So 191.3 is a reasonable answer.

Kite cons. sides

Example 2A: Using Properties of Kites

In kite ABCD, m DAB = 54°, and m CDF = 52°. Find m BCD.

∆BCD is isos. 2 sides isos. ∆

isos. ∆ base s

Def. of s

Polygon Sum Thm.

CBF CDF

m CBF = m CDF

m BCD + m CBF + m CDF = 180°

Example 2A Continued

Substitute m CDF for m CBF.

Substitute 52 for m CDF.

Subtract 104 from both sides.

m BCD + m CDF + m CDF = 180°

m BCD + 52° + 52° = 180°

m BCD = 76°

m BCD + m CBF + m CDF = 180°

A trapezoid is a quadrilateral with exactly one pair of parallel sides. Each of the parallel sides is called a base. The nonparallel sides are called legs. Base angles of a trapezoid are two consecutive angles whose common side is a base.

If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. The following theorems state the properties of an isosceles trapezoid.

Isos. trap. s base

Example 3A: Using Properties of Isosceles

Trapezoids

Find m A.

Same-Side Int. s Thm.

Substitute 100 for m C.

Subtract 100 from both sides.

Def. of s

Substitute 80 for m B

m C + m B = 180°

100 + m B = 180

m B = 80°

A B

m A = m B

m A = 80°

Example 3B: Using Properties of Isosceles

Trapezoids

KB = 21.9 and MF = 32.7.

Find FB.

Isos. trap. s base

Def. of segs.

Substitute 32.7 for FM.

Seg. Add. Post.

Substitute 21.9 for KB and 32.7 for KJ.

Subtract 21.9 from both sides.

KJ = FM

KJ = 32.7

KB + BJ = KJ

21.9 + BJ = 32.7

BJ = 10.8

The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. In Lesson 5-1, you studied the Triangle Midsegment Theorem. The Trapezoid Midsegment Theorem is similar to it.

Example 5: Finding Lengths Using Midsegments

Find EF.

Trap. Midsegment Thm.

Substitute the given values.

Solve.EF = 10.75