Question

a town planner wants to build two new streets, elm street and garden road, to connect parallel streets maple drive and pine avenue. in trapezoid efgh, \\( \overline { e f } \cong \overline { h g } \\). what is the measure of the angle between elm street and pine avenue? \\( 54 ^ { \circ } \\) \\( 72 ^ { \circ } \\) \\( 108 ^ { \circ } \\) \\( 144 ^ { \circ } \\)

Explanation:

Step1: Recall the property of isosceles trapezoid

In an isosceles trapezoid \(EFGH\) (\(\overline{EF}\cong\overline{HG}\)), consecutive - interior angles between the parallel sides (\(FG\parallel EH\)) are supplementary. Let \(\angle FGE = 108^{\circ}\), and we want to find \(\angle FEF\).
The formula for supplementary angles is \(\angle A+\angle B = 180^{\circ}\).

Step2: Calculate the required angle

Let the angle between Elm Street (\(EF\)) and Pine Avenue (\(EH\)) be \(x\).
Since \(FG\parallel EH\) in trapezoid \(EFGH\) and using the supplementary - angle property for consecutive - interior angles in a trapezoid:
\(x + 108^{\circ}=180^{\circ}\)
Subtract \(108^{\circ}\) from both sides of the equation: \(x=180^{\circ}- 108^{\circ}\)

Answer:

\(72^{\circ}\)