Geometry Companion Book, Volume 2

6.2.3 Properties of Kites and Trapezoids (continued) Example 4 Applying Conditions for Isosceles Trapezoids

This is a condition of isosceles trapezoids. If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid.

The values of an unknown, a , in expressions for the base angles of a trapezoid that will make the trapezoid isosceles are determined in this example. It is given that two sides are parallel and the measures of two base angles are given as algebraic expressions with one unknown, a . For the trapezoid to be isosceles, one pair of base angles must be congruent. Set ∠ W ≅ ∠ Z so that m ∠ W = m ∠ Z and substitute the algebraic expressions for the angle measures. This gives 3 a 2 + 20 = 2 a 2 + 45. Solving for a gives a = 5 or a = − 5. The values of an unknown, x , in expressions for the lengths of the diagonals of a trapezoid that will make the trapezoid isosceles are determined in this example. It is given that two sides are parallel and the measures of the diagonals are given as algebraic expressions with one unknown, x . For the trapezoid to be isosceles, the diagonals must be congruent. Set JL ≅ KM so that JL = KM and substitute the algebraic expressions for the lengths. This gives 6 x + 3 = 2 x + 31. Solving for the expression gives x = 7.

Example 5 Finding Lengths Using Midsegments

The midsegment of a trapezoid is parallel to each base and its length is one-half the sum of the lengths of the bases (the average of the base lengths).

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