Question

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

Explanation:

Step1: Recall triangle - angle sum property

In $\triangle ABC$, since $AB = BC$ (equal - length markings), $\angle A=\angle C = 53^{\circ}$. The sum of interior angles of a triangle is $180^{\circ}$. In $\triangle ABC$, $\angle ABC=180^{\circ}-\angle A - \angle C=180^{\circ}-53^{\circ}-53^{\circ}=74^{\circ}$.

Step2: Find the value of $x$

Given $x = 37^{\circ}$.

Step3: Consider $\triangle BDC$

Since $BD = DC$ (equal - length markings), $\angle DBC=\angle C$. We know $\angle C = 53^{\circ}$ and $\angle DBC=x = 37^{\circ}$. In $\triangle BDC$, using the angle - sum property of a triangle ($\angle DBC+\angle C+\angle BDC = 180^{\circ}$), and we want to find $y=\angle BDC$.
$y=180^{\circ}-2\times37^{\circ}=180^{\circ}-74^{\circ}=106^{\circ}$

Answer:

$y = 106$