Lesson 7.5
Properties of Trapezoids and
Kites
Essential Question:
What are the properties of
trapezoids and kites?
Some quadrilaterals
are parallelograms.
But not all… some
quadrilaterals are trapezoids
and some are kites.
Polygons
Quadrilaterals
Rhombuses
Squares
Parallelograms
Rectangles
Trapezoids
Isosceles
Trapezoids
Kites
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4
Trapezoid
• A quadrilateral with exactly one pair of
parallel sides.
A
B
D
C
𝑨𝑩 ∥ 𝑪𝑫
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Trapezoid
• The parallel sides are the BASES.
A
𝑨𝑩 𝒂𝒏𝒅 𝑪𝑫
𝒂𝒓𝒆 𝑩𝑨𝑺𝑬𝑺
B
𝑨𝑪 𝒂𝒏𝒅 𝑩𝑫
𝒂𝒓𝒆 𝑳𝑬𝑮𝑺
C

D
The non-parallel sides are the LEGS.
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Isosceles Trapezoid
• The LEGS (NON-PARALLEL sides) are congruent.
A
B
C
D
𝑨𝑪 ≅ 𝑩𝑫
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Theorem 7.14: Isosceles Trapezoid Base Angles
Theorem
• Each pair of base angles is congruent
∠𝑩 ≅ ∠𝑪
B
A
C
∠𝑨 ≅ ∠𝑫
D
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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.
B
A
C
D
𝐼𝑓 ∠𝑨 ≅ ∠𝑫, 𝑡ℎ𝑒𝑛 𝐴𝐵 ≅ 𝐷𝐶
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Theorem 7.16: Isosceles Trapezoid Diagonals
Theorem
𝐴𝐵 ≅ 𝐷𝐶 𝑖𝑓𝑓 𝐴𝐶 ≅ 𝐵𝐷
A trapezoid is
isosceles if and only
if its diagonals are
congruent.
B
A
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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
3105
2
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• Again, since base angles are equal,
105
2 = ______.
75
75
105
2
105
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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
• Connect the midpoints of the legs.
𝑚𝑖𝑑𝑠𝑒𝑔𝑚𝑒𝑛𝑡
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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
𝐸𝐹 ∥ 𝐷𝐶
1
𝐸𝐹 =
𝐴𝐵 + 𝐷𝐶
2
A
B
F
C
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Example 2
• AB = 24 and DC = 30. Find EF.
A
24
E
D
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EF  (24  30)
2
B
1
 (54)
2
F
30
 27
C
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Your Turn
• AB = 10 and EF = 15. Find CD.
A
E
D
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10
15
?
B
F
C
19
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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This can also be shown as…
Polygons
Quadrilaterals
Rhombuses
Squares
Parallelograms
Rectangles
Trapezoids
Isosceles
Trapezoids
Kites
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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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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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