Warm-Up

• Pg 537-539• 4,10,20,23,43,49

• Answers to evens:• 6.Always• 12. Always • 36. About 6• 42. About 8.3• 48. 2

Use Properties of Trapezoids and Kites

8.5

Trapezoid

• A quadrilateral with exactly one pair of parallel sides, called bases.

Diagonals

• If a trapezoid is isosceles, then each pair of base angles is congruent.

• If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.

• A trapezoid is isosceles if and only if its diagonals are congruent.

EXAMPLE 1 Use a coordinate plane

Show that ORST is a trapezoid.

SOLUTION

Compare the slopes of opposite sides.

Slope of RS =

Slope of OT = 2 – 0 4 – 0 = 2

4 = 12

The slopes of RS and OT are the same, so RS OT .

4 – 3 2 – 0 = 1

2

EXAMPLE 2 Use properties of isosceles trapezoids

Arch

The stone above the arch in the diagram is an isosceles trapezoid. Find m K, m M, and m J.

SOLUTION

STEP 1Find m K. JKLM is an isosceles trapezoid, so K and L are congruent base angles, and m K = m L= 85o.

EXAMPLE 2 Use properties of isosceles trapezoids

STEP 2

Find m M. Because L and M are consecutive interior angles formed by LM intersecting two parallel lines, they are supplementary. So, m M = 180o – 85o = 95o.

STEP 3

Find m J. Because J and M are a pair of base angles, they are congruent, and m J = m M = 95o.

ANSWER

So, m J = 95o, m K = 85o, and m M = 95o.

Midsegment

• The Midsegment is parallel to both bases and half the length of the sum of the bases,

EXAMPLE 3 Use the midsegment of a trapezoid

SOLUTION

Use Theorem 8.17 to find MN.

In the diagram, MN is the midsegment of trapezoid PQRS. Find MN.

MN (PQ + SR)12= Apply Theorem 8.17.

= (12 + 28)12 Substitute 12 for PQ and

28 for XU.

Simplify.= 20

ANSWER The length MN is 20 inches.

GUIDED PRACTICE for Examples 2 and 3

In Exercises 3 and 4, use the diagram of trapezoid EFGH.

3. If EG = FH, is trapezoid EFGH isosceles? Explain.

ANSWER yes, Theorem 8.16

GUIDED PRACTICE for Examples 2 and 3

4. If m HEF = 70o and m FGH = 110o, is trapezoid EFGH isosceles? Explain.

SAMPLE ANSWER Yes;

m EFG = 70° by Consecutive Interior Angles Theorem making EFGH an isosceles trapezoidby Theorem 8.15.

GUIDED PRACTICE for Examples 2 and 3

5. In trapezoid JKLM, J and M are right angles, and JK = 9 cm. The length of the midsegment NP of trapezoid JKLM is 12 cm. Sketch trapezoid

JKLM and its midsegment. Find ML. Explain your reasoning.

J

L

K

M

9 cm

12 cmN P

ANSWER

( 9 + x ) = 121215 cm; Solve for x to find ML.

Kites

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

• Diagonals are perpendicular.

• Exactly one pair of opposite angles are congruent.

EXAMPLE 4 Apply Theorem 8.19

SOLUTION

By Theorem 8.19, DEFG has exactly one pair of congruent opposite angles. Because E G, D and F must be congruent. So, m D = m F. Write and solve an equation to find m D.

Find m D in the kite shown at the right.

m D + m F +124o + 80o = 360o Corollary to Theorem 8.1

m D + m D +124o + 80o = 360o

2(m D) +204o = 360o Combine like terms.

Substitute m D for m F.

Solve for m D. m D = 78o

EXAMPLE 4 Apply Theorem 8.19

GUIDED PRACTICE for Example 4

6. In a kite, the measures of the angles are 3xo, 75o, 90o, and 120o. Find the value of x. What are the

measures of the angles that are congruent?

ANSWER 25; 75o

Classwork

• Pg 546-547

• 4,8,12,14,18,22,26

Homework

• Pg 546-547• 3-27 odd