Trapezoids & Kites

Trapezoids & Kites Objective: Verify and use the properties of trapezoids and kites.

VOCABULARY A trapezoid is a quadrilateral with exactly one pair of parallel sides.

bases and nonparallel sides are called ______. The parallel sides are called _______ legs isosceles trapezoid If the legs are congruent, the trapezoid is an ___________________. In an isosceles trapezoid,  both pairs of base angles are congruent,  the diagonals are congruent. In the isosceles trapezoids at the top of this kettle, each pair of base angles are congruent.

S leg

R

base

T

»

leg

Base angles

U

» base

STUR is an isosceles trapezoid. U , R  _____

SU  ____ TR

T S  ____

SR  TU ____

Trapezoids & Kites Objective: Verify and use the properties of trapezoids and kites.

Median (or Midsegment) of Trapezoids The median (or midsegment) of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases, and its length is half the sum of the lengths of the bases.

median

In trapezoid HJKL, MN =  (HJ + LK)

Trapezoids & Kites Objective: Verify and use the properties of trapezoids and kites.

A kite is a quadrilateral with two pairs of adjacent sides __________ congruent sides and no opposite ______________ congruent.



ST  WT ____ and

WR SR  ____

Theorem: The diagonals of a kite are perpendicular. TR  ___ SW By the HL Theorem, SZT  WZT and SZR  WZR. By CPCTC, STZ  TWZ and SRZ  WRZ.

By the Isosceles  Theorem, TSZ  TWZ and RSZ  RWZ. By the Angle Addition Postulate, TSR  TWR .

Trapezoids & Kites

DNG page 365

Find the measures of the numbered angles in each isosceles trapezoid.

m1 = 180 - 62 m1 = 118

m2 = 62

m1 = 96 m2 = 84

m1 = 99

m1 = 59

m2 = 81

m2 = 121

m1 = 101 m2 =

79

m1 = 67 m2 = 113

DNG page 365

Trapezoids & Kites

ALGEBRA Find the value(s) of the variable(s) in each isosceles trapezoid.

Solve for x : 3x  3 = x + 5

2x = 8 x=4

6x + 20 + 4x = 180 10x + 20 = 180 10x = 160 x = 16

Solve for y : 7x = 2x + 5 y = 6x + 20 5x = 5 y = 6(16) + 20 x=1 y = 116

Trapezoids & Kites

DNG page 365

Find the measures of the numbered angles in each kite.

m1 = 105.5

m1 = 90

m1 = 118

m2 = 105.5

m2 = 25

m2 = 118

Γ

m1 = m2, Let x = m1

m2 = 90  65

x + x + 101 + 48 = 360 2x = 211

m2 = 25°

360  44  80 m1 = 2 m1 = 118°

x = 105.5° m1 = 90

m1 = 90

m1 = 107

m2 = 63

m2 = 107

m3 = 63

m2 = 90 - 27 m2 = 63°

360  59  87 2 m1 = 107°

m1 =

m2 = 51

m3 = 39

m3 = 90 - 51 m3 = 39°

Trapezoids & Kites

DNG page 365

ALGEBRA Find the value(s) of the variable(s) in each kite.

4(28)+13 = 125°

 y

12x = 96 x=8

5x  1 + 8x = 90



Γ

10x  6 + 2x = 90

8x

Solve for x :

5x  15 = 4x + 13 y 9 + y +125 =180

13x = 91 x=7

Solve for y : 2y + 116 =180

x = 28

2y = 64 y = 32

Architecture

6-5 Trapezoids & Kites

Harbour Centre Tower In Vancouver, Canada

The 2nd ring of the ceiling shown at the left is made from isosceles trapezoids that create illusions of circles. What are the measure of base angles of these trapezoids? o Each trapezoid is part of an isosceles △ whose base s are the acute base angles of the trapezoid. o The isosceles△ has a vertex  that is half as large as one of the 20 s at the center of the ceiling. Measure of each angle at the center of the ceiling = 360/20 or 18.

Measure of 1 = 18/2 or 9. Measure of each acute base angle =

180 9 2

 85.5

Measure of each obtuse base angle =180  85.5  94.5

Trapezoids & Kites 2.

1.

PQ =  (WX + ZY)

HJ = 58

12 =  (WX + 19) 24 = WX + 19

ST =  (HJ + LK)

72 =  (HJ + 86) 144 = HJ + 86 -86 =

- 86

-19 =

Substitute. Multiply each side by 2. Subtract 86 from each side.

5 = WX WX = 5

58 = HJ 3.

- 19

4. mQ = mR mQ = 125 A

B

AB =  (60 + 25) AB = 42.5

TU = 28 mE = 95

mS + mR = 180 mS + 126 = 180 mS = 54

mG = 145

Angle Bisector Theorem

2. a. x b. CD c. BC

1. a. a b. FG c. GH

a. Solve for a : FG = GH 2a - 3 = a + 1 a= 4

If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.

b. Solve for FG : FG = 2a - 3 FG = 2(4) - 3 FG = 5 c. Solve for GH : GH = a + 1 GH = 4 + 1 GH = 5

Critical Thinking: Can two angles of a kite be as follows? 1. opposite and acute Yes, the ≅ s can be obtuse example

m1 = 90

3. opposite and complementary Yes, the

≅ s must be 45° or 90°.

m2 = 68

2. consecutive and obtuse Yes, the ≅ s can be obtuse, as well as one other .

4. consecutive and complementary No, if two consecutive s were complementary, then the kite would be concave.