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Warm-Up

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.

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Warm-Up

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  1. 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

  2. Use Properties of Trapezoids and Kites 8.5

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

  4. 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.

  5. Show that ORSTis a trapezoid. = = Slope of RS = 4 – 3 2 – 0 Slope of OT = = 2 – 0 4 – 0 The slopes of RSand OTare the same, so RSOT . 2 1 1 2 4 2 EXAMPLE 1 Use a coordinate plane SOLUTION Compare the slopes of opposite sides.

  6. Arch The stone above the arch in the diagram is an isosceles trapezoid. Find mK, mM, and mJ. STEP 1 Find mK. JKLMis an isosceles trapezoid, so Kand Lare congruent base angles, and mK = mL= 85o. EXAMPLE 2 Use properties of isosceles trapezoids SOLUTION

  7. Find m M. Because Land M are consecutive interior angles formed by LMintersecting two parallel lines, they are supplementary. So, mM= 180o – 85o = 95o. STEP 3 Find mJ. Because J and M are a pair of base angles, they are congruent, and mJ = mM= 95o. ANSWER So, mJ = 95o, m K = 85o, and m M= 95o. EXAMPLE 2 Use properties of isosceles trapezoids STEP 2

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

  9. In the diagram,MNis the midsegment of trapezoidPQRS. FindMN. SOLUTION Use Theorem 8.17 to findMN. 1 =(12+ 28) 2 1 MN (PQ + SR) = 2 ANSWER The length MNis 20 inches. EXAMPLE 3 Use the midsegment of a trapezoid Apply Theorem 8.17. Substitute 12 for PQand 28 for XU. = 20 Simplify.

  10. 3. If EG = FH, is trapezoid EFGHisosceles? Explain. ANSWER yes, Theorem 8.16 for Examples 2 and 3 GUIDED PRACTICE In Exercises 3 and 4, use the diagram of trapezoidEFGH.

  11. 4. If mHEF = 70o and mFGH =110o, is trapezoid EFGHisosceles? Explain. SAMPLE ANSWER Yes; mEFG =70° by Consecutive Interior Angles Theorem making EFGH an isosceles trapezoidby Theorem 8.15. for Examples 2 and 3 GUIDED PRACTICE

  12. J K 9 cm ANSWER 12 cm P N M L 1 2 15 cm; Solve for x to find ML. ( 9 + x ) = 12 for Examples 2 and 3 GUIDED PRACTICE 5. In trapezoid JKLM, Jand M are right angles, and JK = 9 cm. The length of the midsegmentNPof trapezoid JKLMis 12 cm. Sketch trapezoid JKLMand its midsegment. Find ML. Explain your reasoning.

  13. 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.

  14. Find m Din the kite shown at the right. SOLUTION By Theorem 8.19, DEFGhas exactly one pair of congruent opposite angles. Because E G,Dand Fmust be congruent. So, m D = mF. Write and solve an equation to find mD. EXAMPLE 4 Apply Theorem 8.19

  15. mD + m F +124o + 80o = 360o mD + m D +124o + 80o = 360o 2(m D)+204o = 360o m D = 78o Substitute m Dfor m F. Solve for m D. EXAMPLE 4 Apply Theorem 8.19 Corollary to Theorem 8.1 Combine like terms.

  16. 25; 75o ANSWER for Example 4 GUIDED PRACTICE 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?

  17. Classwork • Pg 546-547 • 4,8,12,14,18,22,26

  18. Homework • Pg 546-547 • 3-27 odd

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