Question

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

Explanation:

Step1: Identify congruent triangles

Since $AB = BC$ and $BD$ is common, and $\angle ABD=\angle CBD = 25^{\circ}$, by the Side - Angle - Side (SAS) congruence criterion, $\triangle ABD\cong\triangle CBD$.

Step2: Find angle $x$

In $\triangle ABD$, we know one angle is $50^{\circ}$ and $\angle ABD = 25^{\circ}$. Using the angle - sum property of a triangle ($\angle A+\angle ABD+\angle ADB=180^{\circ}$), and since $\triangle ABD\cong\triangle CBD$, $\angle ADB=\angle CDB$. In $\triangle ABD$, $\angle ADB = 180^{\circ}-50^{\circ}-25^{\circ}=105^{\circ}$. So $x = 105^{\circ}$.

Step3: Find angle $y$

Since $\triangle ABD\cong\triangle CBD$, $\angle C=\angle A = 50^{\circ}$. In $\triangle BCD$, $\angle BDC=x = 105^{\circ}$, and using the angle - sum property of a triangle ($\angle BDC+\angle C+\angle DBC = 180^{\circ}$), we already know $\angle DBC = 25^{\circ}$ and $\angle C = 50^{\circ}$, and $\angle BDC=105^{\circ}$. Also, since $\angle BDC$ and $y$ are vertical angles, $y = 75^{\circ}$ (because $180 - 105=75$).

Answer:

$x = 105^{\circ}, y = 75^{\circ}$