trapezoids and kites chapter 8, section 5 (8.5) essential questions how do i use properties of...
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Trapezoids and Kites
Chapter 8, Section 5
(8.5)
Essential Questions
How do I use properties of trapezoids?
How do I use properties of kites?
VocabularyTrapezoid – a quadrilateral with exactly one pair of parallel sides.
A trapezoid has two pairs of base angles. In this example the base angles are A & B and C & D
A
D C
B
leg leg
base
base
8.14 Base Angles Trapezoid Theorem
If a trapezoid is isosceles, then each pair of base angles is congruent.
A
D C
B
A B, C D
8.15 Base Angles Trapezoid Converse
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. A
D C
B
ABCD is an isosceles trapezoid
8.16 Diagonals of a Trapezoid Theorem
A trapezoid is isosceles if and only if its diagonals are congruent.
A
D C
B
BDAC ifonly and if isosceles is ABCD
Example 1PQRS is an isosceles trapezoid. Find m P, m Q and mR.
50S R
P Q
m R = 50 since base angles are congruent
mP = 130 and mQ = 130 (consecutive angles of parallel lines cut by a transversal are )
Definition
Midsegment of a trapezoid – the segment that connects the midpoints of the legs.
midsegment
8.17 Midsegment Theorem for Trapezoids
The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.
NM
A D
B C
BC) AD2
1 MN ,BC ll MNAD llMN ( ,
DefinitionKite – a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.
8.18 Theorem: Perpendicular Diagonals of a Kite
If a quadrilateral is a kite, then its diagonals are perpendicular.
D
C
A
B
BDAC
8.19 Theorem: Opposite Angles of a Kite
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent
D
C
A
B
A C, B D
Example 2Find the side lengths of the kite.
20
12
1212
UW
Z
Y
X
Example 2 Continued
WX = 4 34
likewise WZ = 4 34
20
12
1212
UW
Z
Y
X
We can use the Pythagorean Theorem to find the side lengths.
122 + 202 = (WX)2
144 + 400 = (WX)2
544 = (WX)2
122 + 122 = (XY)2
144 + 144 = (XY)2
288 = (XY)2
XY =12 2
likewise ZY =12 2
Example 3
Find mG and mJ.
60132
J
G
H K
Since GHJK is a kite G J
So 2(mG) + 132 + 60 = 360
2(mG) =168
mG = 84 and mJ = 84
Try This!RSTU is a kite. Find mR, mS and mT.
x
125
x+30
S
U
R T
x +30 + 125 + 125 + x = 360
2x + 280 = 360
2x = 80
x = 40
So mR = 70, mT = 40 and mS = 125
Homework
Pages 546 (7-15)