Geometry Lesson 6 – 6 Trapezoids and Kites Objective: Apply properties of trapezoids. Apply properties of kites.

GeometryLesson 6 – 6

Trapezoids and Kites

Objective:Apply properties of trapezoids.

Apply properties of kites.

TrapezoidWhat is a trapezoid?

A quadrilateral with exactly one pair of parallel sides.Bases – the parallel sidesLegs – the nonparallel sidesBase angles – the angles formed by the base and

one of the legs

Isosceles trapezoidcongruent legs

Theorem

Theorem 6.21 If a trapezoid is isosceles, then each pair of

base angles is congruent.

Theorem

Theorem 6.22 If a trapezoid has one pair of congruent

base angles, then it is an isosceles trapezoid.

Theorem

Theorem 6.23A trapezoid is isosceles if and only if its

diagonals are congruent.

The speaker shown is an isosceles trapezoid. If m FJH = 85, FK = 8 in. and JG = 19 in. Find each measure.

KH

FGHm

85

8 in19

85

95

95

FH = JGFH = 19FK + KH = 19

8 + KH = 19

KH = 11

XZ

XV

4515 cm

10 cm

XWZm

WXYm

WXYZ isIsosceles trap.

45

135

25 cm

10 cm

Quadrilateral ABCD has vertices A (-3, 4) B (2, 5) C (3, 3) and D (-1, 0). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid.

To be a trapezoid the quad must have one set of parallel sides.Slope of AD = -2 Slope of BC = -2The figure is a trapezoid (AB and DC obviously not parallel)

To be isosceles the diagonals must be congruent. 1.6373433 22 AC

8.5340512 22 BD

Trapezoid ABCD is not an isosceles trapezoid.

Quadrilateral QRST has vertices

Q(8, -4) R(0, 8) S(6, 8) T (-6, -10). Determine whether it is an isosceles trapezoid.

Slope of QR = 3/2 Slope of TS = 3/2

Slope of RS = 0Slope of QT = 3/7

Since one pair of parallel sides QRST is a trapezoid.

2.123721488468 22 QS

0.1910636010860 22 RT

Trapezoid QRST is not an isosceles trapezoid.

Midsegment of a Trapezoid

Midsegment of a trapezoidThe segment that connects the midpoints

of the legs of the trapezoid.

TheoremTrapezoid Midsegment TheoremThe midsegment of a trapezoid is parallel

to each base and its measure is one half the sum of the lengths of the bases.

In the figure segment LH is the midsegment of trapezoid FGJK.

What is the value of x?

KJFGLH 2

1

2.182

115 x (2)(2)

30 = x + 18.2

11.8 = x

Trapezoid ABCD is shown below. If FG II AD, what is the x-coordinate of point G?

G is the midpoint of segment DC.

2

04,

2

201G

2,5.10G

The x-coordinate is 10.5

Kite

A quadrilateral with exactly two pairs of consecutive congruent sides.

TheoremTheorem 6.25 If a quadrilateral is a kite, then its diagonals are

perpendicular.

Where else have learned about the diagonals being perpendicular?Square & Rhombus

Why does this work for all 3 figures? Is a Square and a Rhombus considered a Kite?

TheoremTheorem 6.26 If a quadrilateral is a kite, then exactly one

pair of opposite angles is congruent.

If FGHJ is a kite, find the measure of angle GFJ

Kites have exactly one pair of opposite congruent angles

x x

2x + 128 + 72 = 360

2x + 200 = 3602x = 160

x = 80

80GHJm

If WXYZ is a kite, find ZY.

(PZ)2 + (PY)2 = (ZY)2

82 + 242 = (ZY)2

640 = (ZY)2

ZY640 Remember to simplify!

ZY1064 ZY108

.,5038 ADCmfindBCDmandBADmIf

38

50xx

2x + 38 + 50 = 360 2x + 88 = 360

2x = 272x = 136

136ADCm

If BT = 5 and TC = 8, find CD.

5 8

(BT)2 + (TC)2 = (BC)2

52 + 82 = (BC)2

89 = (BC)2

BC89BC = CD

CD89

Homework

Pg. 440 1 – 7 all, 8 – 24 EOE, 36 – 48 EOE, 66, 70 – 76 E, 82