Question

find the values of x and y.

x = 31 (simplify your answer.)
y = 2 (simplify your answer.)

Explanation:

Step1: Use the property of isosceles triangle

Since \(AB = BC\), \(\triangle ABC\) is isosceles. The base - angles are equal. In \(\triangle ABD\) and \(\triangle CBD\), \(AB = BC\), \(BD = BD\) (common side), and \(AD = DC\) (by the property of the figure). So, \(\triangle ABD\cong\triangle CBD\) (SSS congruence). Then, \(\angle ABD=\angle CBD\).
In \(\triangle ABD\), using the angle - sum property of a triangle (\(\angle A+\angle ABD+\angle ADB = 180^{\circ}\)), and since \(\angle A = 59^{\circ}\) and \(\angle ADB = 90^{\circ}\) (because of the congruence and the nature of the split in the isosceles triangle), \(\angle ABD=180^{\circ}-\angle A - 90^{\circ}=31^{\circ}\). So \(x = 31\).

Step2: Use the property of congruent triangles

Since \(\triangle ABD\cong\triangle CBD\) (SSS: \(AB = BC\), \(BD = BD\), \(AD = DC\)), then \(AD = DC\). Given \(AD = 2\), so \(y=2\).

Answer:

\(x = 31\), \(y = 2\)