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6-6

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Properties of Kites

and Trapezoids

6-6

Warm Up

Lesson Presentation

Lesson Quiz

Holt McDougal Geometry

Warm Up

Solve for x.

1.x2 + 38 = 3x2 – 12

2. 137 + x = 180

3.

4. Find FE.

Objectives

Use properties of kites to solve problems.

Use properties of trapezoids to solve problems.

Vocabulary

kite

trapezoid

base of a trapezoid

leg of a trapezoid

base angle of a trapezoid

isosceles trapezoid

midsegment of a trapezoid

A kiteis 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?

1

Make a Plan

Understand the Problem

The diagonals of a kite are perpendicular, so the four triangles are right triangles. Let N represent the intersection of the diagonals. Use the Pythagorean Theorem and the properties of kites to find , and . Add these lengths to find the length of .

2

Example 1 Continued

The answer will be the amount of wood Lucy has left after cutting the dowel.

3

Solve

Example 1 Continued

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.

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.

4

Example 1 Continued

Look Back

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?

1

Understand the Problem

Check It Out! Example 1 Continued

- The answer has two parts.
- • the total length of binding Daryl needs
- • the number of packages of binding Daryl must buy

Make a Plan

2

Check It Out! Example 1 Continued

The diagonals of a kite are perpendicular, so the four triangles are right triangles. Use the Pythagorean Theorem and the properties of kites to find the unknown side lengths. Add these lengths to find the perimeter of the kite.

3

Solve

perimeter of PQRS =

Check It Out! Example 1 Continued

Pyth. Thm.

Pyth. Thm.

packages of binding

Check It Out! Example 1 Continued

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.

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.

4

Check It Out! Example 1 Continued

Look Back

Example 2A: Using Properties of Kites

In kite ABCD, mDAB = 54°, and mCDF = 52°. Find mBCD.

Kite cons. sides

∆BCD is isos.

2 sides isos. ∆

isos. ∆base s

CBF CDF

mCBF = mCDF

Def. of s

Polygon Sum Thm.

mBCD + mCBF + mCDF = 180°

Example 2A Continued

mBCD + mCBF + mCDF = 180°

Substitute mCDF for mCBF.

mBCD + mCDF+ mCDF = 180°

Substitute 52 for mCDF.

mBCD + 52°+ 52° = 180°

Subtract 104 from both sides.

mBCD = 76°

Example 2B: Using Properties of Kites

In kite ABCD, mDAB = 54°, and mCDF = 52°. Find mABC.

ADC ABC

Kite one pair opp. s

Def. of s

mADC = mABC

Polygon Sum Thm.

mABC + mBCD + mADC + mDAB = 360°

Substitute mABC for mADC.

mABC + mBCD + mABC+ mDAB = 360°

Example 2B Continued

mABC + mBCD + mABC + mDAB = 360°

mABC + 76°+ mABC + 54° = 360°

Substitute.

2mABC = 230°

Simplify.

mABC = 115°

Solve.

Example 2C: Using Properties of Kites

In kite ABCD, mDAB = 54°, and mCDF = 52°. Find mFDA.

CDA ABC

Kite one pair opp. s

mCDA = mABC

Def. of s

mCDF + mFDA = mABC

Add. Post.

52° + mFDA = 115°

Substitute.

mFDA = 63°

Solve.

Check It Out! Example 2a

In kite PQRS, mPQR = 78°, and mTRS = 59°. Find mQRT.

Kite cons. sides

∆PQR is isos.

2 sides isos. ∆

RPQ PRQ

isos. ∆base s

mQPT = mQRT

Def. of s

Check It Out! Example 2a Continued

mPQR + mQRP + mQPR = 180°

Polygon Sum Thm.

Substitute 78 for mPQR.

78° + mQRT+ mQPT = 180°

78° + mQRT + mQRT = 180°

Substitute.

78° + 2mQRT = 180°

Substitute.

Subtract 78 from both sides.

2mQRT = 102°

mQRT = 51°

Divide by 2.

Check It Out! Example 2b

In kite PQRS, mPQR = 78°, and mTRS = 59°. Find mQPS.

QPS QRS

Kite one pair opp. s

Add. Post.

mQPS = mQRT + mTRS

mQPS = mQRT + 59°

Substitute.

mQPS = 51° + 59°

Substitute.

mQPS = 110°

Check It Out! Example 2c

In kite PQRS, mPQR = 78°, and mTRS = 59°. Find each mPSR.

Polygon Sum Thm.

mSPT + mTRS + mRSP = 180°

mSPT = mTRS

Def. of s

mTRS + mTRS + mRSP = 180°

Substitute.

59° + 59°+ mRSP = 180°

Substitute.

Simplify.

mRSP = 62°

A trapezoidis 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 anglesof 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.

Reading Math

Theorem 6-6-5 is a biconditional statement. So it is true both “forward” and “backward.”

Example 3A: Using Properties of Isosceles Trapezoids

Find mA.

mC + mB = 180°

Same-Side Int. s Thm.

100 + mB = 180

Substitute 100 for mC.

mB = 80°

Subtract 100 from both sides.

A B

Isos. trap. s base

mA = mB

Def. of s

mA = 80°

Substitute 80 for mB

Example 3B: Using Properties of Isosceles Trapezoids

KB = 21.9 and MF = 32.7.

Find FB.

Isos. trap. s base

KJ = FM

Def. of segs.

KJ = 32.7

Substitute 32.7 for FM.

Seg. Add. Post.

KB + BJ = KJ

Substitute 21.9 for KB and 32.7 for KJ.

21.9 + BJ = 32.7

BJ = 10.8

Subtract 21.9 from both sides.

Example 3B Continued

Same line.

KFJ MJF

Isos. trap. s base

Isos. trap. legs

SAS

∆FKJ ∆JMF

CPCTC

BKF BMJ

Vert. s

FBK JBM

Example 3B Continued

Isos. trap. legs

AAS

∆FBK ∆JBM

CPCTC

FB = JB

Def. of segs.

FB = 10.8

Substitute 10.8 for JB.

Check It Out! Example 3a

Find mF.

mF + mE = 180°

Same-Side Int. s Thm.

E H

Isos. trap. s base

mE = mH

Def. of s

mF + 49°= 180°

Substitute 49 for mE.

mF = 131°

Simplify.

Check It Out! Example 3b

JN = 10.6, and NL = 14.8. Find KM.

Isos. trap. s base

Def. of segs.

KM = JL

JL = JN + NL

Segment Add Postulate

KM = JN + NL

Substitute.

KM = 10.6 + 14.8 = 25.4

Substitute and simplify.

Example 4A: Applying Conditions for Isosceles Trapezoids

Find the value of a so that PQRS is isosceles.

Trap. with pair base s isosc. trap.

S P

mS = mP

Def. of s

Substitute 2a2 – 54 for mS and a2 + 27 for mP.

2a2 – 54 = a2 + 27

Subtract a2 from both sides and add 54 to both sides.

a2 = 81

a = 9 or a = –9

Find the square root of both sides.

Example 4B: Applying Conditions for Isosceles Trapezoids

AD = 12x – 11, and BC = 9x – 2. Find the value of x so that ABCD is isosceles.

Diags. isosc. trap.

Def. of segs.

AD = BC

Substitute 12x – 11 for AD and 9x – 2 for BC.

12x – 11 = 9x – 2

Subtract 9x from both sides and add 11 to both sides.

3x = 9

x = 3

Divide both sides by 3.

Check It Out! Example 4

Find the value of x so that PQST is isosceles.

Trap. with pair base s isosc. trap.

Q S

mQ = mS

Def. of s

Substitute 2x2 + 19 for mQ and 4x2 – 13 for mS.

2x2 + 19 = 4x2 – 13

Subtract 2x2 and add 13 to both sides.

32 = 2x2

Divide by 2 and simplify.

x = 4 or x = –4

The midsegment of a trapezoidis 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

1

16.5 = (25 + EH)

2

Check It Out! Example 5

Find EH.

Trap. Midsegment Thm.

Substitute the given values.

Simplify.

Multiply both sides by 2.

33= 25 + EH

Subtract 25 from both sides.

13= EH

Lesson Quiz: Part I

1. Erin is making a kite based on the pattern below. About how much binding does Erin need to cover the edges of the kite?

In kite HJKL, mKLP = 72°,

and mHJP = 49.5°. Find each

measure.

2. mLHJ3. mPKL

Lesson Quiz: Part II

Use the diagram for Items 4 and 5.

4. mWZY = 61°. Find mWXY.

5.XV = 4.6, and WY = 14.2. Find VZ.

6. Find LP.