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Properties of Trapezoids and Kites in Geometry

Learn about the properties of trapezoids and kites in geometry, including theorems, midsegments, diagonals, and angles. Solve problems and find the value of x in rectangles and the perimeter of rhombuses.

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Properties of Trapezoids and Kites in Geometry

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  1. DRILL • If the two diagonals of a rectangle are 4x + 10 and 2x + 36, then what is the value of x? • If two adjacent sides of a rhombus are 3x – 4 and 6x – 19, then what is the perimeter of the rhombus?

  2. 9.4 Trapezoids and Kites Geometry Mr. Calise

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

  4. A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. A trapezoid has two pairs of base angles. For instance in trapezoid ABCD D and C are one pair of base angles. The other pair is A and B. The nonparallel sides are the legs of the trapezoid. Using properties of trapezoids

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

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

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

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

  9. PQRS is an isosceles trapezoid. Find mP, mQ, mR. PQRS is an isosceles trapezoid, so mR = mS = 50°. Because S and P are consecutive interior angles formed by parallel lines, they are supplementary. So mP = 180°- 50° = 130°, and mQ = mP = 130° Ex. 1: Using properties of Isosceles Trapezoids 50° You could also add 50 and 50, get 100 and subtract it from 360°. This would leave you 260/2 or 130°.

  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. Use the midsegment theorem for trapezoids. DG = ½(EF + CH)= ½ (8 + 20) = 14” Ex. 3: Finding Midsegment lengths of trapezoids E F D G D C

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

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

  18. WXYZ is a kite so the diagonals are perpendicular. You can use the Pythagorean Theorem to find the side lengths. WX = √202 + 122≈ 23.32 XY = √122 + 122≈ 16.97 Because WXYZ is a kite, WZ = WX ≈ 23.32, and ZY = XY ≈ 16.97 Ex. 4: Using the diagonals of a kite

  19. Ex. 5: Angles of a kite • Find mG and mJ in the diagram at the right. SOLUTION: GHJK is a kite, so G ≅ J and mG = mJ. 2(mG) + 132° + 60° = 360°Sum of measures of int. s of a quad. is 360° 2(mG) = 168°Simplify mG = 84° Divide each side by 2. So, mJ = mG = 84° 132° 60°

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