Exercise 13 Page 612

The triangle on the left has a base angle that is $y^{\circ }.$ Second, the center triangle has one angle that is $x^{\circ }.$

Isosceles Triangle

According to the Base Angles Theorem, if two sides of a triangle are congruent, then the angles opposite them are congruent. Therefore, the second base angle of the left triangle is also $y^{\circ }.$

Equilateral Triangle

According to the Corollary to the Base Angles Theorem, if a triangle is equilateral, then it is equiangular. Therefore, we know that the two unknown angles are $x^{\circ }$ as well.

Finding $x$ and $y$

According to the Triangle Sum Theorem, the sum of the measures of the interior angles of a triangle is $180^{\circ }.$ Therefore, we can write the following equation. Let's replace $x^{\circ }$ with its measure in our diagram.

Note that two sides are parallel. If we view the right leg of the left triangle as a transversal, we can identify a pair of alternate interior angles.

According to the Alternate Interior Angles Theorem, if two parallel lines are cut by a transversal, then the pair of alternate interior angles are congruent. Therefore, we know that $y^{\circ }=60^{\circ }.$