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Trapezoids

Trapezoids. 5-5. Show that ORST is a trapezoid. =. =. Slope of RS =. 4 – 3. 2 – 0. Slope of OT =. =. 2 – 0. 4 – 0. The slopes of RS and OT are the same, so RS OT. 2. 1. 1. 2. 4. 2. EXAMPLE 1. Use a coordinate plane. SOLUTION. Compare the slopes of opposite sides. –2.

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Trapezoids

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  1. Trapezoids 5-5

  2. 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.

  3. –2 –1 = = Slope of ST = 2 , which is undefined = Slope of OR = The slopes of ST and ORare not the same, soST is not parallel to OR . 2 – 4 3 – 0 ANSWER 0 – 0 4 – 2 Because quadrilateral ORST has exactly one pair of parallel sides, it is a trapezoid. 3 0 EXAMPLE 1 Use a coordinate plane

  4. = = Slope of RS = 2 – 0 5 – 3 Slope of OT = 4 – 0 4 – 0 The slopes of RSand OTare the same, so RSOT . 1 1 2 2 for Example 1 GUIDED PRACTICE 1. What If?In Example 1, suppose the coordinates of point Sare (4, 5). What type of quadrilateral is ORST? Explain. SOLUTION Compare the slopes of opposite sides.

  5. –3 undefined = Slope of ST = 0 , undefined = Slope of OR = The slopes of ST and ORare the same, soST is parallel to OR . 3 – 0 2 – 5 ANSWER Parallelogram; opposite pairs of sides are parallel. 0 – 0 4 – 4 3 0 for Example 1 GUIDED PRACTICE

  6. 2. In Example 1, which of the interior angles of quadrilateral ORSTare supplementary angles? Explain your reasoning. ANSWER O and R , T and S are supplementary angles, as RS and OR are parallel lines cut by transversals OR and ST, therefore the pairs of consecutive interior angles are supplementary by theorem 8.5 for Example 1 GUIDED PRACTICE

  7. 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

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

  9. 3x + 75 + 90 + 120 = 360° 3x + 285 = 360° 3x = 75 x = 25 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? SOLUTION STEP 1 Sum of the angles in a quadrilateral = 360° Combine like terms Subtract Divided by 3 from each side

  10. 3x ANSWER The value of x is 25 and the measures of the angles that are congruent is 75 = 3 25 for Example 4 GUIDED PRACTICE STEP 2 Substitute = 75 Simplify

  11. 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

  12. STEP 2 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

  13. 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.

  14. 3. If EG = FH, is trapezoid EFGHisosceles? Explain. ANSWER Yes, trapezoid EFGH is isosceles, if and only if its diagonals are congruent. As, it is given its diagonals are congruent, therefore by theorem 8.16 the trapezoid is isosceles. for Examples 2 and 3 GUIDED PRACTICE In Exercises 3 and 4, use the diagram of trapezoidEFGH.

  15. 4. If mHEF = 70o and mFGH =110o, is trapezoid EFGHisosceles?Explain. ANSWER G and F are consecutive interior angles as EF HG. Because FGH=110°, therefore EFG=70° as they are supplementary angles. BHG=110°, as the sum of a quadrilateral is 360° . The base angles are congruent that is, 110° each therefore, the trapezoid is isosceles by theorem 8.15. for Examples 2 and 3 GUIDED PRACTICE

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

  17. 1 (JK + ML) NP = 2 12 24 = 9 + x 15 = x 1 = (9 + x) 2 for Examples 2 and 3 GUIDED PRACTICE Apply Theorem 8.17. Substitute Multiply each side by 2 Simplify.

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