Properties and conditions 6-6 Kites and Trapezoids Properties and conditions

Kites A kite is a quadrilateral with exactly two pairs of congruent consecutive sides.

Properties of Kites

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.

Isosceles Trapezoids If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. The following theorems state the properties of an isosceles trapezoid.

Properties of Isosceles Trapezoids

Midsegments of Trapezoids The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. In Lesson 5-1, you studied the Triangle Midsegment Theorem. The Trapezoid Midsegment Theorem is similar to it.

Trapezoid Midsegment Theorem

Lets apply! Example 1 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.

Lets apply! Example 1 Continued mBCD + mCBF + mCDF = 180° mBCD + mCDF + mCDF = 180° Substitute mCDF for mCBF. mBCD + 52° + 52° = 180° Substitute 52 for mCDF. mBCD = 76° Subtract 104 from both sides.

Lets apply! Example 2 In kite ABCD, mDAB = 54°, and mCDF = 52°. Find mFDA. CDA  ABC Kite  one pair opp. s  mCDA = mABC Def. of  s mCDF + mFDA = mABC  Add. Post. 52° + mFDA = 115° Substitute. mFDA = 63° Solve.

Example 3: Applying Conditions for Isosceles Trapezoids Lets apply! Example 3: Applying Conditions for Isosceles Trapezoids Find the value of a so that PQRS is isosceles. S  P Trap. with pair base s   isosc. trap. mS = mP Def. of  s 2a2 – 54 = a2 + 27 Substitute 2a2 – 54 for mS and a2 + 27 for mP. a2 = 81 Subtract a2 from both sides and add 54 to both sides. a = 9 or a = –9 Find the square root of both sides.

Example 4: Applying Conditions for Isosceles Trapezoids Lets apply! Example 4: Applying Conditions for Isosceles Trapezoids AD = 12x – 11, and BC = 9x – 2. Find the value of x so that ABCD is isosceles. Diags.   isosc. trap. AD = BC Def. of  segs. 12x – 11 = 9x – 2 Substitute 12x – 11 for AD and 9x – 2 for BC. 3x = 9 Subtract 9x from both sides and add 11 to both sides. x = 3 Divide both sides by 3.

Example 5 Conditions of Midsegments Lets apply! Example 5 Conditions of Midsegments Find EH. Trap. Midsegment Thm. 1 16.5 = (25 + EH) 2 Substitute the given values. Simplify. 33 = 25 + EH Multiply both sides by 2. 13 = EH Subtract 25 from both sides.