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8.5 Trapezoids and Kites

8.5 Trapezoids and Kites. Objectives:. Use properties of trapezoids. Use properties of kites. A trapezoid is a quadrilateral with exactly one pair of parallel sides. Using properties of trapezoids. If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.

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8.5 Trapezoids and Kites

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  1. 8.5 Trapezoids and Kites

  2. Objectives: • Use properties of trapezoids. • Use properties of kites.

  3. A trapezoid is a quadrilateral with exactly one pair of parallel sides. Using properties of trapezoids

  4. If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. Using properties of trapezoids

  5. Theorem 6.14 If a trapezoid is isosceles, then each pair of base angles is congruent. A ≅ B, C ≅ D Trapezoid Theorems

  6. Theorem 6.15 If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. ABCD is an isosceles trapezoid Trapezoid Theorems

  7. Theorem 6.16 A trapezoid is isosceles if and only if its diagonals are congruent. ABCD is isosceles if and only if AC ≅ BD. Trapezoid Theorems

  8. PQRS is an isosceles trapezoid. Find mP, mQ, mR. Ex. 1: Using properties of Isosceles Trapezoids 50°

  9. Ex.2: The stone above the arch in the diagram is an isosceles trapezoid. Find

  10. Show that ABCD is a trapezoid. Compare the slopes of opposite sides. The slope of AB = 5 – 0 = 5 = - 1 0 – 5 -5 The slope of CD = 4 – 7 = -3 = - 1 7 – 4 3 The slopes of AB and CD are equal, so AB ║ CD. The slope of BC = 7 – 5 = 2 = 1 4 – 0 4 2 The slope of AD = 4 – 0 = 4 = 2 7 – 5 2 The slopes of BC and AD are not equal, so BC is not parallel to AD. So, because AB ║ CD and BC is not parallel to AD, ABCD is a trapezoid. Ex. 2: Using properties of trapezoids

  11. The midsegment of a trapezoid is the segment that connects the midpoints of its legs. Theorem 6.17 is similar to the Midsegment Theorem for triangles. Midsegment of a trapezoid

  12. The midsegment of a trapezoid is parallel to each base and its length is one half the sums of the lengths of the bases. MN║AD, MN║BC MN = ½ (AD + BC) Theorem 6.17: Midsegment of a trapezoid

  13. LAYER CAKE A baker is making a cake like the one at the right. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 20 inches. How big should the middle layer be? Ex. 3: Finding Midsegment lengths of trapezoids

  14. Ex.4: In the diagram, is the midsegment of trapezoid PQRS. Find MN. Ex.5: In the diagram, is the midsegment of trapezoid DEFG. Find HK.

  15. A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. Using properties of kites

  16. Theorem 6.18 If a quadrilateral is a kite, then its diagonals are perpendicular. AC  BD Kite theorems

  17. Theorem 6.19 If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent. A ≅ C, B ≅ D Kite theorems

  18. Ex.7: Find in the kite shown below. Ex.8: 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.

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