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6.5 Trapezoids and Kites

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If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. ... A trapezoid is isosceles if and only if its diagonals are congruent. ... – PowerPoint PPT presentation

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Title: 6.5 Trapezoids and Kites


1
6.5 Trapezoids and Kites
  • Geometry
  • Mrs. Spitz
  • Spring 2005

2
Objectives
  • Use properties of trapezoids.
  • Use properties of kites.

3
Assignment
  • pp. 359-360 2-33

4
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.

5
Using properties of trapezoids
  • If the legs of a trapezoid are congruent, then
    the trapezoid is an isosceles trapezoid.

6
Trapezoid Theorems
  • Theorem 6.14
  • If a trapezoid is isosceles, then each pair of
    base angles is congruent.
  • ?A ? ?B, ?C ? ?D

7
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

8
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.

9
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.
10
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.

11
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.

12
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)

13
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?

14
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
15
Using properties of kites
  • A kite is a quadrilateral that has two pairs of
    consecutive congruent sides, but opposite sides
    are not congruent.

16
Kite theorems
  • Theorem 6.18
  • If a quadrilateral is a kite, then its diagonals
    are perpendicular.
  • AC ? BD

17
Kite theorems
  • Theorem 6.19
  • If a quadrilateral is a kite, then exactly one
    pair of opposite angles is congruent.
  • ?A ? ?C, ?B ? ?D

18
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

19
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
20
Reminder
  • Quiz after this section
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