Chapter Properties of kites and trapezoids

Objectives Use properties of kites to solve problems. Use properties of trapezoids to solve problems.

Kite  A kite is a quadrilateral with exactly two pairs of congruent consecutive sides.

Properties

Problem solving application  Lucy is framing a kite with wooden dowels. She uses two dowels that measure 18 cm, one dowel that measures 30 cm, and two dowels that measure 27 cm. To complete the kite, she needs a dowel to place along. She has a dowel that is 36 cm long. About how much wood will she have left after cutting the last dowel?

Lucy’s Kite

SOlution  The answer will be the amount of wood Lucy has left after cutting the dowel. 1 Understand the Problem 2 Make a Plan

 The diagonals of a kite are perpendicular, so the four triangles are right triangles. Let N represent the intersection of the diagonals. Use the Pythagorean Theorem and the properties of kites to find, and. Add these lengths to find the length of.

Solve 3

solution Lucy needs to cut the dowel to be 32.4 cm long. The amount of wood that will remain after the cut is, 36 – 32.4  3.6 cm Lucy will have 3.6 cm of wood left over after the cut.

Example 2A: Using Properties of Kites  In kite ABCD, m  DAB = 54°, and m  CDF = 52°. Find m  BCD.

Solution   ADC   ABC Kite  one pair opp.  s   m  ADC = m  ABC Def. of   s 

Trapezoid  A trapezoid is a quadrilateral with exactly one pair of parallel sides. Each of the parallel sides is called a base. The nonparallel sides are called legs. Base angles of a trapezoid are two consecutive angles whose common side is a base.

Isosceles trapezod  If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. The following theorems state the properties of an isosceles trapezoid.

Trapezoids

Example #1 Find m  A. m  A = 80°

Example#2  KB = 21.9 and MF =  Find FB. FB = 10.8

Example#3  Find m  F. m  F = 131°

Example#4  Find the value of a so that PQRS is isosceles. a = 9 or a = –9

Example  AD = 12 x – 11, and BC = 9 x – 2. Find the value of x so that ABCD is isosceles. x = 3

Midsegment  The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. In Lesson 5-1, you studied the Triangle Midsegment Theorem. The Trapezoid Midsegment Theorem is similar to it.

Conditions for midsegment

example  Find EF EF = 10.75

example  Find EH 13 = EH