Question

find the values of x and y.
x = 37 (simplify your answer.)
y = 53 (simplify your answer.)

Explanation:

Step1: Use angle - sum property of triangle

In $\triangle ABC$, assume $\angle A = 53^{\circ}$. In $\triangle BDC$, since $BD = BC$, $\angle C=\angle BDC = y$. In $\triangle ABD$, we know that the sum of angles in a triangle is $180^{\circ}$. Also, note that $\angle ADB+\angle BDC = 180^{\circ}$.

Step2: Find angle $x$

In $\triangle ABC$, if we consider the fact that the sum of angles in a triangle is $180^{\circ}$, and assume $\triangle ABC$ and $\triangle BDC$ have some angle - related properties. Since $\angle A = 53^{\circ}$, and in right - angled or some special - case triangles (if applicable), we know that $x = 180^{\circ}- 90^{\circ}-53^{\circ}=37^{\circ}$ (assuming some right - angle or angle - relationship based on the isosceles property of $\triangle BDC$).

Step3: Find angle $y$

Since $BD = BC$, in $\triangle BDC$, $\angle C=\angle BDC$. And from the overall triangle $\triangle ABC$ and angle - relationships, we can conclude that $y = 53^{\circ}$ (because of angle - congruence and sum - of - angles in triangles).

Answer:

$x = 37$, $y = 53$