Question

defg is an isosceles trapezoid. find the measure of $\angle e$..
e f
75°
105°
d g
$m\angle e = 95°$
$m\angle e = 75°$
$m\angle e = 85°$
$m\angle e = 105°$

Explanation:

Step1: Recall Isosceles Trapezoid Angles

In an isosceles trapezoid, consecutive angles between the bases are supplementary, and base angles are equal. Also, angles adjacent to each non - parallel side (leg) are supplementary. Here, \(EF\parallel DG\), so \(\angle E\) and \(\angle D\) are supplementary? Wait, no. Wait, in trapezoid \(DEFG\), \(DE\) and \(FG\) are the legs, \(EF\) and \(DG\) are the bases. So \(\angle E\) and \(\angle F\) are adjacent to base \(EF\), and \(\angle D\) and \(\angle G\) are adjacent to base \(DG\). Also, in an isosceles trapezoid, angles on the same leg are supplementary. Wait, actually, in an isosceles trapezoid, each pair of base angles is equal, and consecutive angles between the bases are supplementary. Wait, let's look at the sides: \(E - F - G - D - E\). So \(EF\parallel DG\), so \(\angle E\) and \(\angle D\) are same - side interior angles, so they should be supplementary? Wait, no, \(DE\) is a leg, so the sides are \(EF\) (top base), \(FG\) (leg), \(DG\) (bottom base), \(DE\) (leg). So \(EF\parallel DG\), so the transversal \(DE\) creates same - side interior angles \(\angle E\) and \(\angle D\), which should be supplementary? Wait, but also, in an isosceles trapezoid, base angles are equal. Wait, maybe a better way: in an isosceles trapezoid, angles adjacent to a leg are supplementary. So \(\angle E\) and \(\angle D\): are they adjacent to leg \(DE\)? Yes, \(\angle E\) is at \(E\) between \(EF\) and \(DE\), \(\angle D\) is at \(D\) between \(DE\) and \(DG\). Since \(EF\parallel DG\), \(\angle E+\angle D = 180^{\circ}\)? Wait, but \(\angle D = 105^{\circ}\), then \(\angle E=180 - 105=75^{\circ}\)? Wait, no, wait the other way: or \(\angle E\) and \(\angle F\): \(\angle F = 75^{\circ}\), and in an isosceles trapezoid, are \(\angle E\) and \(\angle D\) related? Wait, maybe I made a mistake. Wait, let's recall the properties of isosceles trapezoid: 1. The base angles are equal. 2. Consecutive angles between the bases are supplementary. So if \(EF\parallel DG\), then \(\angle E+\angle D = 180^{\circ}\) and \(\angle F+\angle G = 180^{\circ}\), and \(\angle E=\angle F\)? No, wait no. Wait, base angles: the angles adjacent to each base are equal. So for base \(EF\), the angles at \(E\) and \(F\) are base angles? No, base angles are the angles adjacent to each of the non - parallel sides (legs). Wait, maybe the correct property is that in an isosceles trapezoid, each pair of angles adjacent to a leg is supplementary. So \(\angle E\) and \(\angle D\) are adjacent to leg \(DE\), so \(\angle E+\angle D = 180^{\circ}\)? But \(\angle D = 105^{\circ}\), then \(\angle E=180 - 105 = 75^{\circ}\)? Wait, but also, \(\angle F = 75^{\circ}\), and in an isosceles trapezoid, are \(\angle E\) and \(\angle F\) equal? No, that can't be. Wait, maybe I mixed up the bases. Let's look at the diagram: \(E\) and \(F\) are on the top base \(EF\), \(D\) and \(G\) are on the bottom base \(DG\). So the legs are \(DE\) and \(FG\). So the angles at \(E\) and \(F\) are the top base angles, and angles at \(D\) and \(G\) are the bottom base angles. In an isosceles trapezoid, top base angles are equal, bottom base angles are equal, and a top base angle and a bottom base angle (adjacent to the same leg) are supplementary. So \(\angle E\) (top base angle) and \(\angle D\) (bottom base angle) are adjacent to leg \(DE\), so \(\angle E+\angle D=180^{\circ}\)? But \(\angle D = 105^{\circ}\), so \(\angle E = 180 - 105=75^{\circ}\). Also, \(\angle F = 75^{\circ}\) (given), which matches the top base angles being equal.

Step2: Check the Options

We found that \(\angle E = 180^{\circ}-\angle D\). Since \(\angle D = 105^{\circ}\), \(\angle E=180 - 105 = 75^{\circ}\). Looking at the options, the option with \(m\angle E = 75^{\circ}\) is one of the choices.

Answer:

\(m\angle E = 75^{\circ}\) (the option with \(m\angle E = 75^{\circ}\))