Question

1. find the measure of ∠3.

118°
62°
118°
31°

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of interior angles of a triangle is 180°. In the lower - triangle where ∠3 is located, assume the other non - given angle is part of a pair of vertical angles. The angle opposite the 118° angle in the upper - triangle has the same measure of 118° (vertical angles are equal). Let's consider the lower triangle.

Step2: Calculate ∠3

Let the measure of ∠3 be \(x\). In the lower triangle, if we assume the other non - given angle is \(y\) and we know that the sum of the interior angles of a triangle is 180°. The angle opposite the 118° angle in the upper triangle (equal to 118°) and the two angles in the lower triangle satisfy the angle - sum property. Since the two triangles are symmetric about the diagonal, we can also note that the two non - 118° angles in the whole rhombus are equal. The sum of the two non - 118° angles in the whole rhombus is \(360^{\circ}- 2\times118^{\circ}=360^{\circ}-236^{\circ} = 124^{\circ}\), so each of these non - 118° angles (including the one related to ∠3) is \(\frac{124^{\circ}}{2}=62^{\circ}\). In the lower triangle, if we consider the fact that the angle opposite the 118° angle in the upper triangle is 118°, and the sum of angles in the lower triangle is 180°. Let the measure of ∠3 be \(x\). We know that \(x=\frac{180^{\circ}- 118^{\circ}}{2}\).
\[x = 31^{\circ}\]

Answer:

31°