6.5 Trapezoids and Kites

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6.5 Trapezoids and Kites

• Geometry
• Mrs. Spitz
• Spring 2005

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Objectives

• Use properties of trapezoids.
• Use properties of kites.

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Assignment

• pp. 359-360 2-33

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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.

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Using properties of trapezoids

• If the legs of a trapezoid are congruent, then
  the trapezoid is an isosceles trapezoid.

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Trapezoid Theorems

• Theorem 6.14
• If a trapezoid is isosceles, then each pair of
  base angles is congruent.
• ?A ? ?B, ?C ? ?D

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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

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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.

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Ex. 1 Using properties of Isosceles Trapezoids

• PQRS is an isosceles trapezoid. Find m?P, m?Q,
  m?R.
• PQRS is an isosceles trapezoid, so m?R m?S
  50. Because ?S and ?P are consecutive interior
  angles formed by parallel lines, they are
  supplementary. So m?P 180- 50 130, and
  m?Q m?P 130

50
You could also add 50 and 50, get 100 and
subtract it from 360. This would leave you
260/2 or 130.
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Ex. 2 Using properties of trapezoids

• 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.

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Midsegment of a trapezoid

• The midsegment of a trapezoid is the segment that
  connects the midpoints of its legs. Theorem 6.17
  is similar to the Midsegment Theorem for
  triangles.

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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 of the bases.
• MNAD, MNBC
• MN ½ (AD BC)

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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?

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Ex. 3 Finding Midsegment lengths of trapezoids
E

• Use the midsegment theorem for trapezoids.
• DG ½(EF CH)
• ½ (8 20) 14

F
D
G
D
C
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Using properties of kites

• A kite is a quadrilateral that has two pairs of
  consecutive congruent sides, but opposite sides
  are not congruent.

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Kite theorems

• Theorem 6.18
• If a quadrilateral is a kite, then its diagonals
  are perpendicular.
• AC ? BD

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Kite theorems

• Theorem 6.19
• If a quadrilateral is a kite, then exactly one
  pair of opposite angles is congruent.
• ?A ? ?C, ?B ? ?D

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Ex. 4 Using the diagonals of a kite

• WXYZ is a kite so the diagonals are
  perpendicular. You can use the Pythagorean
  Theorem to find the side lengths.
• WX v202 122 23.32
• XY v122 122 16.97
• Because WXYZ is a kite, WZ WX 23.32, and ZY
  XY 16.97

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Ex. 5 Angles of a kite

• Find m?G and m?J
• in the diagram at the
• right.
• SOLUTION
• GHJK is a kite, so ?G ? ?J and m?G m?J.
• 2(m?G) 132 60 360Sum of measures of int.
  ?s of a quad. is 360
• 2(m?G) 168Simplify
• m?G 84 Divide
  each side by 2.
• So, m?J m?G 84

132
60
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Reminder

• Quiz after this section