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Chapter 6 Section 5

Chapter 6 Section 5. Trapezoids. Warm-Up. Using P for parallelogram, R for rectangle, S for square, and Rh for rhombus, write the letters of all the quadrilaterals that have these properties . 1) Perpendicular diagonals S, Rh 2) Diagonals that bisect each other P, R, S, Rh

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Chapter 6 Section 5

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  1. Chapter 6Section 5 Trapezoids

  2. Warm-Up • Using P for parallelogram, R for rectangle, S for square, and Rh for rhombus, write the letters of all the quadrilaterals that have these properties. • 1) Perpendicular diagonals • S, Rh • 2) Diagonals that bisect each other • P, R, S, Rh • 3) Congruent diagonals • R, S • 4) Two pairs of congruent opposite sides •  P, R, S, Rh • 5) Supplementary consecutive angles • P, R, S, Rh • 6) Diagonals that bisect opposite angles • S, Rh • 7) Four right angles • R, S • 8) Two pairs of opposite parallel sides •  P, R, S, Rh

  3. Vocabulary Trapezoid- A quadrilateral with exactly one pair of parallel sides. Bases-The parallel sides of a trapezoid. Legs-The nonparallel sides of a trapezoid. Base Angles-The angles are on same parallel line. Isosceles Trapezoid- If the legs of a trapezoid are congruent then is it an isosceles trapezoid. Base Leg Base angles Leg Base angles Base

  4. Vocabulary cont. Base Theorem 6-14-Both pairs of base angles of an isosceles trapezoid are congruent. Theorem 6-15- The diagonals of an isosceles trapezoid are congruent. Median of a trapezoid-The segment that joins the midpoints of the legs. Theorem 6-16- The median of a trapezoid is parallel to the bases, and its measure is ½ the sum of the measures of the bases. Median = ½(b1+ b2) Leg Leg Base b1 Median b2

  5. Example 1: ABCD is an isosceles trapezoid. Decide whether each statement is true or false. Explain. A)AC = BD True; Diagonals of an isosceles trapezoid are congruent. B) AD is congruent to CB True; Legs of an isosceles trapezoid are congruent. C) CA and BD bisect each other. False; If the diagonals bisected each other it would be a parallelogram not a trapezoid. A B E D C

  6. Example 2: Given trapezoid EZOI with median AB, find the value of x. Median = ½(b1+ b2) AB = ½(EZ + IO) 13 = ½(4x – 10 + 3x + 8) 13 = ½(7x - 2) 26 = 7x – 2 28 = 7x 4 = x 4x - 10 E Z 13 A B I O 3x + 8

  7. Example 3: Given trapezoid EZOI with median AB, find the value of x. Median = ½(b1+ b2) AB = ½(EZ + IO) 10 = ½(3x – 1 + 7x + 1) 10 = ½(10x) 10 = 5x 2 = x 3x - 1 E Z 10 A B I O 7x + 1

  8. Example 4: WXYZ is an isosceles trapezoid with median MN. Use the given information to solve each problem. A) Find MN, if WZ = 11 and XY = 3 Median = ½(b1+ b2) MN = ½(XY + WZ) MN = ½(3 + 11) MN = ½(14) MN = 7 B) Find m<XMN if m<WZN = 78. The base angles of an isosceles trapezoid are congruent. m<WZN = m<ZWM 78 = m<ZWM <XMN and <ZWN are corresponding angles so they are congruent. m<ZWM = m<XMN 78 = m<XMN X Y M N W Z C) If MN = 10 and WZ = 14, find XY. Median = ½(b1+ b2) MN = ½(XY + WZ) 10 = ½(XY + 14) 20 = XY + 14 6 = XY

  9. Example 5: WXYZ is an isosceles trapezoid with median MN. Use the given information to solve each problem. A) What is the value of x if m<MWZ = 15x – 5 and m<WZN = 90 – 4x? The base angles of an isosceles trapezoid are congruent. m<WZN = m<MWZ 90 – 4x = 15x - 5 90 = 19x - 5 95 = 19x 5 = x B) If m<XWZ = 2x – 7 and m<XYZ = 117, find the value of x.. The base angles of an isosceles trapezoid are congruent. m<XWZ = m<WZY Consecutive angles are supplementary. 180 = m<XYZ + m<WZY 180 = 2x – 7 + 117 180 = 2x + 110 70 = 2x 35 = x X Y M N W Z C) If MN = 10x + 3, WZ = 11, and XY = 8x + 19. Find the value of x. Median = ½(b1+ b2) MN = ½(XY + WZ) 10x + 3 = ½(8x + 19 + 11) 10x + 3 = ½(8x + 30) 10x + 3 = 4x + 15 6x + 3 = 15 6x = 12 x = 2

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