Transcript 6.5 Trapezoids and Kites Geometry Mrs. Spitz Spring 2005 Objectives: Use properties of trapezoids.  Use properties of kites. 

6.5 Trapezoids and Kites
Geometry
Mrs. Spitz
Spring 2005
Objectives:
Use properties of trapezoids.
 Use properties of kites.

Assignment:

pp. 359-360 #2-33
Using properties of trapezoids

A trapezoid is a
quadrilateral with exactly
one pair of parallel
sides. The parallel sides
are the bases. A
trapezoid has two pairs
of base angles. For
instance in trapezoid
ABCD D and C are
one pair of base angles.
The other pair is A and
B. The nonparallel
sides are the legs of the
trapezoid.
A
base
leg
leg
D
B
base
C
Using properties of trapezoids

If the legs of a
trapezoid are
congruent, then the
trapezoid is an
isosceles trapezoid.
Trapezoid Theorems
Theorem 6.14
 If a trapezoid is
isosceles, then each
pair of base angles
is congruent.
 A ≅ B, C ≅ D
A
D
B
C
Trapezoid Theorems
Theorem 6.15
 If a trapezoid has a
pair of congruent
base angles, then it
is an isosceles
trapezoid.
 ABCD is an
isosceles trapezoid
A
D
B
C
Trapezoid Theorems
Theorem 6.16
 A trapezoid is
isosceles if and only
if its diagonals are
congruent.
 ABCD is isosceles if
and only if AC ≅ BD.
A
D
B
C
Ex. 1: Using properties of Isosceles Trapezoids


PQRS is an isosceles
trapezoid. Find mP, m RQ = 2.16 cm
mQ, mR.
m PS = 2.16 cm
PQRS is an isosceles
S
R
trapezoid, so mR =
50°
mS = 50°. Because S
and P are consecutive
interior angles formed by
Q
P
parallel lines, they are
You could also add 50 and 50,
supplementary. So
get 100 and subtract it from
mP = 180°- 50° = 130°,
360°. This would leave you
and mQ = mP = 130°
260/2 or 130°.
Ex. 2: Using properties of trapezoids
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Show that ABCD is a trapezoid.
Compare the slopes of opposite sides.
 The slope of AB = 5 – 0 = 5 = - 1
0 – 5 -5
 The slope of CD = 4 – 7 = -3 = - 1
7–4 3
The slopes of AB and CD are equal, so AB ║
CD.
 The slope of BC = 7 – 5 = 2 = 1
4–0 4 2
 The slope of AD = 4 – 0 = 4 = 2
7–5 2
The slopes of BC and AD are not equal, so
BC is not parallel to AD.
So, because AB ║ CD and BC is not parallel to
AD, ABCD is a trapezoid.
8
C(4, 7)
6
B(0, 5)
4
D(7, 4)
2
5
A(5, 0)
Midsegment of a trapezoid

The midsegment of a
trapezoid is the
B
segment that
connects the
midpoints of its legs.
Theorem 6.17 is
similar to the
A
Midsegment
Theorem for
triangles.
C
midsegment
D
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
A
of the bases.
 MN║AD, MN║BC
 MN = ½ (AD + BC)
B
M
C
N
D
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
E

Use the midsegment
theorem for
trapezoids.
 DG = ½(EF + CH)=
½ (8 + 20) = 14”
D
C
F
G
D
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
B
diagonals are
perpendicular.
 AC  BD
C
D
A
Kite theorems
Theorem 6.19
 If a quadrilateral is a
kite, then exactly one
pair of opposite
B
angles is congruent.
 A ≅ C, B ≅ D
C
D
A
Ex. 4: Using the diagonals of a kite
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WXYZ is a kite so
the diagonals are
perpendicular. You
can use the
Pythagorean
Theorem to find the
side lengths.
X
12
W
WX = √202 + 122 ≈ 23.32
XY = √122 + 122 ≈ 16.97
Because WXYZ is a kite, WZ
= WX ≈ 23.32, and ZY = XY ≈
16.97
20
U 12
12
Z
Y
Ex. 5: Angles of a kite
J

Find mG and mJ
in the diagram at the
H 132°
60°
K
right.
SOLUTION:
G
GHJK is a kite, so G ≅ J and mG = mJ.
2(mG) + 132° + 60° = 360°Sum of measures of int. s of a quad. is 360°
2(mG) = 168°Simplify
mG = 84° Divide each side by 2.
So, mJ = mG = 84°
Reminder:

Quiz after this section