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, ef ≅ hg. what is the measure of the angle between elm street and pine avenue? 54° 72° 108° 144°

Explanation:

Step1: Recall the property of isosceles trapezoid

In an isosceles trapezoid (since \(EF\cong HG\), trapezoid \(EFGH\) is isosceles), consecutive - angles between the non - parallel sides are supplementary. Let \(\angle FGH = 108^{\circ}\). The sum of adjacent angles in a trapezoid (with the two bases parallel) is \(180^{\circ}\).

Step2: Calculate the required angle

Let the angle between Elm Street (\(EF\)) and Pine Avenue (\(EH\)) be \(x\). In an isosceles trapezoid \(EFGH\) with \(FG\parallel EH\), we know that \(\angle FGH+\angle GHE = 180^{\circ}\) (supplementary angles as \(FG\parallel EH\)). Also, \(\angle FGH\) and \(\angle GHE\) are related to the angles of the trapezoid. Another property: the base - angles of an isosceles trapezoid are equal. The sum of the interior angles of a trapezoid is \(360^{\circ}\), and for an isosceles trapezoid \(EFGH\) with \(FG\parallel EH\), \(\angle F=\angle GHE\) and \(\angle E=\angle FGH\). But if we use the property of supplementary angles for the parallel lines \(FG\) and \(EH\) cut by the transversal \(GH\) (or \(EF\)). The angle adjacent to \(108^{\circ}\) (in terms of the parallel lines \(FG\) and \(EH\)): \(x = 180 - 108\).

Answer:

\(72^{\circ}\)