Trapezoids
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
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Printed: February 26, 2015
AUTHORS
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
www.ck12.org
C HAPTER
Chapter 1. Trapezoids
1
Trapezoids
Here you’ll learn what a trapezoid is and what properties it possesses.
What if you were told that the polygonABCD is an isosceles trapezoid and that one of its base angles measures 38◦ ?
What can you conclude about its other base angle? After completing this Concept, you’ll be able to find the value
of a trapezoid’s unknown angles and sides.
Watch This
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/136715
CK-12 Trapezoids
Guidance
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
An isosceles trapezoid is a trapezoid where the non-parallel sides are congruent.
The base angles of an isosceles trapezoid are congruent. If ABCD is an isosceles trapezoid, then 6 A ∼
= 6 B and
∼
6 C = 6 D.
1
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The converse is also true. If a trapezoid has congruent base angles, then it is an isosceles trapezoid. The diagonals
of an isosceles trapezoid are also congruent. The midsegment (of a trapezoid) is a line segment that connects the
midpoints of the non-parallel sides:
There is only one midsegment in a trapezoid. It will be parallel to the bases because it is located halfway between
them.
Midsegment Theorem: The length of the midsegment of a trapezoid is the average of the lengths of the bases.
If EF is the midsegment, then EF =
AB+CD
.
2
Example A
Look at trapezoid T RAP below. What is m6 A?
T RAP is an isosceles trapezoid. m6 R = 115◦ also.
To find m6 A, set up an equation.
2
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Chapter 1. Trapezoids
115◦ + 115◦ + m6 A + m6 P = 360◦
230◦ + 2m6 A = 360◦
→ m6 A = m6 P
2m6 A = 130◦
m6 A = 65◦
Notice that m6 R+m6 A = 115◦ +65◦ = 180◦ . These angles will always be supplementary because of the Consecutive
Interior Angles Theorem.
Example B
Is ZOID an isosceles trapezoid? How do you know?
40◦ 6= 35◦ , ZOID is not an isosceles trapezoid.
Example C
Find x. All figures are trapezoids with the midsegment marked as indicated.
a)
b)
c)
3
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Answer:
a) x is the average of 12 and 26.
12+26
2
=
38
2
= 19
b) 24 is the average of x and 35.
x + 35
= 24
2
x + 35 = 48
x = 13
c) 20 is the average of 5x − 15 and 2x − 8.
5x − 15 + 2x − 8
= 20
2
7x − 23 = 40
7x = 63
x=9
MEDIA
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CK-12 Trapezoids
Guided Practice
T RAP an isosceles trapezoid.
Find:
1.
2.
3.
4.
4
m6
m6
m6
m6
T PA
PT R
PZA
ZRA
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Chapter 1. Trapezoids
Answers:
1. 6 T PZ ∼
= 6 RAZ so m6 T PA = 20◦ + 35◦ = 55◦ .
2. 6 T PA is supplementary with 6 PT R, so m6 PT R = 125◦ .
3. By the Triangle Sum Theorem, 35◦ + 35◦ + m6 PZA = 180◦ , so m6 PZA = 110◦ .
4. Since m6 PZA = 110◦ , m6 RZA = 70◦ because they form a linear pair. By the Triangle Sum Theorem, m6 ZRA =
90◦ .
Explore More
1. Can the parallel sides of a trapezoid be congruent? Why or why not?
For questions 2-8, find the length of the midsegment or missing side.
2.
3.
4.
5.
6.
7.
Find the value of the missing variable(s).
5
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8.
Find the lengths of the diagonals of the trapezoids below to determine if it is isosceles.
9. A(−3, 2), B(1, 3),C(3, −1), D(−4, −2)
10. A(−3, 3), B(2, −2),C(−6, −6), D(−7, 1)
6