Warm Up
Lesson 7.5
Properties of Trapezoids and
Kites
Essential Question:
What are the properties of
trapezoids and kites?
What you will learn
• Use properties of trapezoids
• Use the Trapezoid Midsegment Theorem to
find distances
• Use properties of kites
• Identify quadrilaterals
Quadrilaterals
• Some
quadrilaterals
are
parallelograms
.
Quadrilaterals
Parallelograms
Rhombuses
Rectangles
Squares
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But not all… Some quadrilaterals are Trapezoids.
And some quadrilaterals are Kites.
Quadrilaterals
Kites
Parallelograms
Rhombuses
Trapezoids
Rectangles
Squares
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Trapezoid
• A quadrilateral with exactly one pair of
parallel sides.
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Trapezoid
• The parallel sides are the BASES.

The non-parallel sides are the LEGS.
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These are trapezoids:
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Isosceles Trapezoid
• The legs are congruent.
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Theorem 7.14: Isosceles Trapezoid Base Angles
Theorem
• Each pair of base angles is congruent
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Theorem 7.15: Isosceles Trapezoid Base Angles
Converse
If a trapezoid has one pair of congruent
base angles, then the trapezoid is isosceles.
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Theorem 7.16: Isosceles Trapezoid Diagonals
Theorem
A trapezoid is
isosceles if and only
if its diagonals are
congruent.
B
A
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AC  BD
C
D
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Example 1
• Find the measure of 1, 2, and 3 if the
figure is an isosceles trapezoid.
1
75
3
2
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Solution
• In an isosceles trapezoid, base
75
angles are congruent. 1 is _______.
751
75
3
2
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• The bases are parallel, so 3 = _____.
105
75
75
105
3
2
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• Again, since base angles are equal,
105
2 = ______.
75
75
105
2
105
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Your Turn
Name the vertices of CDEF.
• C must be where?
• Now label the rest.
• CDEF is an isosceles trapezoid with CE = 10
and E = 95. Find DF, C, D and F.
D
C
?
95
95
85
?
E
85
?
F
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Your Turn (cont.)
• CE is a diagonal and is 10.
10 (In an isosc. trap., diags are .)
• DF = ____.
D
C
95 95
85
E
85
F
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Triangle Review
• The midsegment of a
triangle connects the
midpoints of two
sides.
• The midsegment is
parallel to the third
side and one-half its
length.
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Trapezoid Midsegment
• Connects the midpoints of the legs.
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Theorem 7.17: Trapezoid Midsegment Theorem
• The midsegment of
a trapezoid is
parallel to each
base and its length
is one-half the sum
of the bases. (it is
the average of the
two bases.
E
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D
EF AB
A
EF DC
1
EF   AB  DC 
2
B
F
C
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Example 2
• AB = 24 and DC = 30. Find EF.
1
EF  (24  30)
2
1
 (54)
2
E
A
24
B
F
 27
D
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C
23
Your Turn
• AB = 10 and EF = 15. Find CD.
A
E
D
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10
15
?
B
F
C
24
A
Solution
E
D
1
EF   AB  CD 
2
1
15  10  CD 
2
30  10  CD
CD  20
10
B
15
F
?
20
C
Now for the easy way…
The average is right in the middle
of two numbers. So think…
20
10 → 15 → ____
+5
+5
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Trapezoid Summary
• A trapezoid has 2 parallel sides.
• A trapezoid is NOT a parallelogram.
• The legs of an isosceles trapezoid are congruent.
• The base angles of an isosceles trapezoid are
congruent.
• The midsegment of a trapezoid is one-half the sum of
the two bases.
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Kites
A kite is a quadrilateral with two pairs of
congruent sides, but opposite sides are
not congruent. (They’re not parallel,
either.)
These are
And these are
congruent
congruent
consecutive
consecutive
sides. sides.
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Kites
A kite is a quadrilateral with two pairs of
congruent sides, but opposite sides are
not parallel.
There are NO
parallel sides.
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Kite Properties
• The diagonals of a kite are perpendicular.
(Theorem 7.18: Kite Diagonals Theorem)
• One pair of opposite angles is congruent.
(Theorem 7.19: Kite Opposite Angles
Theorem)
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Example 3
• GHJK is a kite. Find GH.
H
G
J
5
2
K
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Use Pythagorean
Theorem.
Solution
Segments
GK and GH
are
congruent.
52 + 22 = GK2
H
29= GK2
29
G
J
5
29
2
K
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Your turn
• RSTU is a kite. Find R, S, T.
S
R x + 30
x
T
125
U
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S
?
125
Solution
R x + 30
70
The sum of the angles in a
quadrilateral is 360.
40 x
T
125
U
x + 30 + 125 + x + 125 = 360
2x + 280 = 360
2x = 80
x = 40
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True or False?
• In ABCD, diagonals AC and BD are perpendicular.
• ABCD is a kite.
• True or False?
• FALSE.
• Why?
• The diagonals of a rhombus are also perpendicular.
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Quadrilaterals
Quadrilateral
Kite
Parallelogram
Rhombus
Rectangle
Trapezoid
Isosceles
Trapezoid
Square
Each shape has all the
properties of all of the shapes
above it.
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Example 4
Which of these have at least one pair of
congruent sides?
Quadrilateral
Kite



Parallelogram
Rhombus
Trapezoid
Isosceles
Trapezoid


Rectangle

Square
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Example 5
Which of these have at least one pair of
congruent angles?
Quadrilateral
Kite



Parallelogram
Rhombus
Trapezoid
Isosceles
Trapezoid


Rectangle

Square
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Example 6
Which of these have two pairs of congruent
angles?
Quadrilateral

Parallelogram
Kite

Rhombus
Trapezoid
Isosceles
Trapezoid


Rectangle

Square
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Example 7
In which of these figures are diagonals
perpendicular?
Quadrilateral
Kite


Parallelogram
Rhombus
Trapezoid
Isosceles
Trapezoid
Rectangle

Square
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Example 8
In which of these is the sum of the angles 360?
Quadrilateral



Kite


Parallelogram
Rhombus
Trapezoid
Isosceles
Trapezoid


Rectangle

All of these have angle
Square
sums of 360 -- they are
all quadrilaterals.
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What is this?
What we know:
•Diagonals
bisect each
other.
•Diagonals
congruent.
Must be a…
Rectangle.
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What is this?
What we know:
•Diagonals
bisect each
other.
•Diagonals
congruent.
•Diagonals
perpendicular.
Must be a…
Square.
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What is this?
What we know:
•One pair of
opposite sides
parallel.
•Base angles
congruent.
•Must be an…
Isosceles
Trapezoid.
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What is this?
What we know:
•Both pair of
opposite angles
congruent.
•And that’s all.
•Must be a…
Parallelogram.
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Why?
1 + 2 + 3 + 4 = 360
1
1  3; 2  4
2
2 1 + 2 2 = 360
1 + 2 = 180
4
3
1 and 2 are
supplementary.
The same is true for all
consecutive pairs.
Opposite sides are parallel.
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What is this?
What we know:
•Four angles
congruent
•Must be a…
Rectangle.
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What kinds of quadrilaterals meet the
conditions shown?
1. Rectangle
2. Square
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In the following examples, which
two segments or angles must be
congruent to enable you to prove
ABCD is the given quadrilateral?
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Example 9: show ABCD is
a rectangle.
Possible Solutions:
Which two segments or
angles must be congruent
to enable you to prove
ABCD is a rectangle?
A
B
D
C
Show A  B
B  C
C  D
D  A
or AC  BD
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Example 10: show ABCD
is a parallelogram.
Possible Solutions:
Which two segments
or angles must be
congruent to enable
you to prove ABCD is a
parallelogram?
A
B
AD  BC
A C & B  D
D
C
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Your Turn. Show trapezoid ABCD is an
Isosceles Trapezoid.
Possible Answers
AD  BC
A
B
A  B
D  C
D
C
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What kind of quadrilateral has at least one
pair of opposite sides congruent?
• Parallelogram
• Rhombus
• Rectangle
• Square
• Isosceles Trapezoid
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