Post on 04-Jan-2016
transcript
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