Question

solving problems with trapezoids
city planners want to design a park between parallel streets, main
street and willow lane, in the shape of a trapezoid. there are two
paths of equal length on the east and west sides of the park. the
border of the park makes a 60° angle between willow lane and the
east path.
what is the angle between main street and the west path?
120
what is the angle between the west path and willow lane?

Explanation:

Step1: Recall properties of isosceles trapezoid

Since the two paths (legs of the trapezoid) are equal in length, the trapezoid is isosceles. In an isosceles trapezoid, consecutive angles between the bases are supplementary. That is, if one angle between a base (Willow Lane) and a leg (east path) is \(60^{\circ}\), and we consider the relationship between angles formed by the parallel lines (Main Street and Willow Lane) and the non - parallel sides (paths).

Step2: Use the supplementary angle property

Let \(x\) be the angle between the west path and Willow Lane. We know that for a trapezoid with parallel bases (Main Street and Willow Lane), the sum of adjacent angles (one on each base and between a base and a leg) is \(180^{\circ}\). But also, in an isosceles trapezoid, the angles adjacent to each non - parallel side (path) are equal. The angle between Willow Lane and the east path is \(60^{\circ}\), and the trapezoid is isosceles. So the angle between the west path and Willow Lane is \(60^{\circ}\) (because in an isosceles trapezoid, the base - angles adjacent to each non - parallel side are equal).

Answer:

\(60\)