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Trapezoids and Kites
1/16/13Mrs. B
Objectives:
Use properties of trapezoids.Use properties of kites.
Using properties of trapezoids
A trapezoid is a quadrilateral with exactly one pair of parallel sides called bases.
A trapezoid has two pairs of base angles. Ex. D and C
And A and B. The nonparallel sides
are the legs of the trapezoid.
base
base
legleg
A B
D C
Using properties of trapezoids
If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.
Isosceles Trapezoid
If a trapezoid is isosceles, then each pair of base angles is congruent.A ≅ B, C ≅ D
A B
D C
Isosceles Trapezoid
If a trapezoid is isosceles, then adjacent angles (not bases) are supplementary.<A + <D = 180<B + <C = 80
A B
D C
Ex. 1: Using properties of Isosceles Trapezoids
Given, angle X is 50Find <R, < P and <Q,
m PS = 2.16 cm
m RQ = 2.16 cm
S R
P Q
50°
Isosceles Trapezoid
A trapezoid is isosceles if and only if its diagonals are congruent.
ABCD is isosceles if and only if AC BD.≅
A B
D C
Midsegment of a trapezoid
The midsegment of a trapezoid is the segment that connects the midpoints of its legs.
midsegment
B C
DA
Theorem 6.17: Midsegment of a trapezoid
The midsegment of a trapezoid is parallel to each base and its length is one half the sums of the lengths of the bases.
MN║AD, MN║BC MN = ½ (AD + BC)
NM
A D
CB
Ex. 3: Finding Midsegment lengths of trapezoids
LAYER CAKE A baker is making a cake like the one at the right. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 20 inches. How big should the middle layer be?
Ex. 3: Finding Midsegment lengths of trapezoids
Use the midsegment theorem for trapezoids.
DG = ½(EF + CH)=½ (8 + 20) = 14” C
D
E
D
G
F
Using properties of kites
A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.
Kite theorems
Theorem 6.18 If a quadrilateral is a
kite, then its diagonals are perpendicular.
AC BD
B
CA
D
Kite theorems
Theorem 6.19 If a quadrilateral is a
kite, then exactly one pair of opposite angles is congruent.
A ≅ C B not =D
B
CA
D
Ex. 4: Using the diagonals of a kite
WXYZ is a kite so the diagonals are perpendicular. You can use the Pythagorean Theorem to find the side lengths.
WX = XY =
12
1220
12
U
X
Z
W Y
Ex. 5: Angles of a kite
Find mG and mJ
in the diagram.
J
G
H K132° 60°