Exercise 12 Page 379

The given parallelogram is a rhombus. Let's find the measure of each numbered angle one at a time.

Measure of $\angle 1$ and $\angle 4$

To help with the process of solving, we will label the angle next to $\angle 1$ as $\angle 1^{\prime }.$

Recall that in a rhombus, each diagonal bisects a pair of opposite angles. Therefore, $\angle 1$ and $\angle 1^{\prime }$ are congruent angles. Also, notice that $\angle 1$ and $\angle 4$ are alternate interior angles. Since a rhombus is a parallelogram, we know that its opposite sides are parallel. Therefore, by the Alternate Interior Angles Theorem, $\angle 1$ and $\angle 4$ are congruent.