6 6 Properties of Kites and Trapezoids Warm

6 -6 Properties of Kites and Trapezoids Warm Up Solve for x. 1. x 2 + 38 = 3 x 2 – 12 5 or – 5 2. 137 + x = 180 43 3. 156 4. Find FE. Holt Geometry

6 -6 Properties of Kites and Trapezoids 6 -6 Holt Geometry Properties of Kites and Trapezoids

6 -6 Properties of Kites and Trapezoids A kite is a quadrilateral with exactly two pairs of congruent consecutive sides. Holt Geometry

6 -6 Properties of Kites and Trapezoids Holt Geometry

6 -6 Properties of Kites and Trapezoids Example 2 A: Using Properties of Kites In kite ABCD, m DAB = 54°, and m CDF = 52°. Find m BCD. Kite cons. sides ∆BCD is isos. 2 sides isos. ∆ CBF CDF isos. ∆ base s m CBF = m CDF Def. of s m BCD + m CBF + m CDF = 180° Polygon Sum Thm. Holt Geometry

6 -6 Properties of Kites and Trapezoids Example 2 A Continued m BCD + m CBF + m CDF = 180° Substitute m CDF m BCD + m CBF + m CDF = 180° for m CBF. m BCD + 52° = 180° m BCD = 76° Holt Geometry Substitute 52 for m CBF. Subtract 104 from both sides.

6 -6 Properties of Kites and Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. Each of the parallel sides is called a base. The nonparallel sides are called legs. Base angles of a trapezoid are two consecutive angles whose common side is a base. Holt Geometry

6 -6 Properties of Kites and Trapezoids If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. Holt Geometry

6 -6 Properties of Kites and Trapezoids Holt Geometry

6 -6 Properties of Kites and Trapezoids Example 3 A: Using Properties of Isosceles Trapezoids Find m A. m C + m B = 180° 100 + m B = 180 Holt Geometry Same-Side Int. s Thm. Substitute 100 for m C. m B = 80° A B Subtract 100 from both sides. Isos. trap. s base m A = m B Def. of s m A = 80° Substitute 80 for m B

6 -6 Properties of Kites and Trapezoids Example 3 B: Using Properties of Isosceles Trapezoids KB = 21. 9 m and MF = 32. 7. Find FB. Isos. trap. s base KJ = FM Def. of segs. KJ = 32. 7 Substitute 32. 7 for FM. KB + BJ = KJ Seg. Add. Post. 21. 9 + BJ = 32. 7 Substitute 21. 9 for KB and 32. 7 for KJ. BJ = 10. 8 Subtract 21. 9 from both sides. Holt Geometry

6 -6 Properties of Kites and Trapezoids Check It Out! Example 3 b JN = 10. 6, and NL = 14. 8. Find KM. Isos. trap. s base KM = JL JL = JN + NL Def. of segs. KM = JN + NL Substitute. Segment Add Postulate KM = 10. 6 + 14. 8 = 25. 4 Substitute and simplify. Holt Geometry

6 -6 Properties of Kites and Trapezoids Example 4 A: Applying Conditions for Isosceles Trapezoids Find the value of a so that PQRS is isosceles. Trap. with pair base s isosc. trap. S P m S = m P 2 a 2 – 54 = a 2 Substitute 2 a 2 – 54 for m S and + 27 2 a + 27 for m P. = 81 a = 9 or a = – 9 Holt Geometry Def. of s Subtract a 2 from both sides and add 54 to both sides. Find the square root of both sides.

6 -6 Properties of Kites and Trapezoids Example 4 B: Applying Conditions for Isosceles Trapezoids AD = 12 x – 11, and BC = 9 x – 2. Find the value of x so that ABCD is isosceles. Diags. isosc. trap. AD = BC Def. of segs. Substitute 12 x – 11 for AD and 12 x – 11 = 9 x – 2 for BC. 3 x = 9 x=3 Holt Geometry Subtract 9 x from both sides and add 11 to both sides. Divide both sides by 3.

6 -6 Properties of Kites and Trapezoids The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. Holt Geometry

6 -6 Properties of Kites and Trapezoids Holt Geometry

6 -6 Properties of Kites and Trapezoids Example 5: Finding Lengths Using Midsegments Find EF. Trap. Midsegment Thm. Substitute the given values. EF = 10. 75 Holt Geometry Solve.

6 -6 Properties of Kites and Trapezoids Check It Out! Example 5 Find EH. Trap. Midsegment Thm. 1 16. 5 = 2 (25 + EH) Substitute the given values. Simplify. 33 = 25 + EH Multiply both sides by 2. 13 = EH Subtract 25 from both sides. Holt Geometry

6 -6 Properties of Kites and Trapezoids Lesson Quiz: Part I 1. Erin is making a kite based on the pattern below. About how much binding does Erin need to cover the edges of the kite? about 191. 2 in. In kite HJKL, m KLP = 72°, and m HJP = 49. 5°. Find each measure. 2. m LHJ Holt Geometry 81° 3. m PKL 18°

6 -6 Properties of Kites and Trapezoids Lesson Quiz: Part II Use the diagram for Items 4 and 5. 4. m WZY = 61°. Find m WXY. 119° 5. XV = 4. 6, and WY = 14. 2. Find VZ. 9. 6 6. Find LP. 18 Holt Geometry