8.6 Trapezoids

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8.6 Trapezoids
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Objectives

• Recognize and apply properties of trapezoids
• Solve problems using the medians of trapezoids

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Trapezoids

• A trapezoid is a quadrilateral with exactly one
  pair of parallel sides.
• The parallel sides are called the bases.
• A trapezoid has two pairs of base angles. In
  trapezoid ABCD, ?D and ?C are one pair of base
  angles. The other pair is ?A and ?B.
• The nonparallel sides of the trapezoid are called
  the legs.

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Trapezoids

• If the legs of a trapezoid are ?, then it is
  called an isosceles trapezoid.

Theorem 8.18 Both pairs of base ?s of an
isosceles trapezoid are ?.(?A ? ?B and ?C ? ?D)
Theorem 8.19 The diagonals of an isosceles
trapezoid are ?.(AC ? BD)
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Example 1
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Example 1
Proof
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Your Turn
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Your Turn
Proof
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Example 2
The top of this work station appears to be two
adjacent trapezoids. Determine if they are
isosceles trapezoids.
Each pair of base angles is congruent, so the
legs are the same length.
Answer Both trapezoids are isosceles.
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Your Turn
The sides of a picture frame appear to be two
adjacent trapezoids. Determine if they are
isosceles trapezoids.
Answer yes
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Example 3a
ABCD is a quadrilateral with vertices A(5, 1),
B(3, 1), C(2, 3), and D(2, 4). Verify that
ABCD is a trapezoid.
A quadrilateral is a trapezoid if exactly one
pair of opposite sides are parallel. Use the
Slope Formula.
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Example 3a
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Example 3b
ABCD is a quadrilateral with vertices A(5, 1),
B(3, 1), C(2, 3), and D(2, 4). Determine
whether ABCD is an isosceles trapezoid. Explain.
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Example 3b
First use the Distance Formula to show that the
legs are congruent.
Answer Since the legs are not congruent, ABCD is
not an isosceles trapezoid.
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Your Turn
QRST is a quadrilateral with vertices Q(3, 2),
R(2, 2), S(1, 4), and T(6, 4).
a. Verify that QRST is a trapezoid.
Answer Exactly one pair of opposite sides is
parallel. Therefore, QRST is a trapezoid.
b. Determine whether QRST is an isosceles
trapezoid. Explain.
Answer Since the legs are not congruent, QRST
is not an isosceles trapezoid.
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Medians of Trapezoids

• The segment that joins the midpoints of the legs
  of a trapezoid is called the median (MN). It is
  also referred to as the midsegment.
• Theorem 8.20 The median of a trapezoid is to
  the bases and its measure is ½ the sum of the
  measures of the bases.
• BC AD
• MN ½ (BC AD)

median
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Example 4a
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Example 4a
Theorem 8.20
Substitution
Multiply each side by 2.
Subtract 20 from each side.
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Example 4b
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Example 4b
Consecutive Interior Angles Theorem
Substitution
Combine like terms.
Divide each side by 9.
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Your Turn
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Assignment

• Pre-AP Geometry Pg. 442 9 19, 22 28, 32,
  34
• Geometry Pg. 442 9 18, 22 - 28