Question

find the values of x and y.

x = (simplify your answer.)

Explanation:

Step1: Use the property of isosceles triangles

Since \(AB = BC\) and \(AD = DC\), \(\triangle ABD\cong\triangle CBD\) (by SSS - Side - Side - Side congruence criterion). So, \(\angle ABD=\angle CBD\).
In \(\triangle ABD\), using the angle - sum property of a triangle (\(\angle A+\angle ABD+\angle ADB = 180^{\circ}\)), but we can also note that for \(\angle ABD\), given \(\angle A = 51^{\circ}\) and \(\triangle ABD\) is part of the congruent triangles.
We know that in \(\triangle ABD\), \(\angle ABD\) can be found as follows:
Since \(\triangle ABD\) and \(\triangle CBD\) are congruent and \(AB = BC\), \(AD = DC\), \(BD=BD\).
The angle \(\angle ABD\) is calculated as \(x = 51^{\circ}\) (because \(\triangle ABD\cong\triangle CBD\) and corresponding angles of congruent triangles are equal).

Answer:

\(x = 51\)