Chapter 12, Problem 14CR

To determine

To find: The values of $x$ and $y$ in the given diagram.

Answer to Problem 14CR

The values of $x$ and $y$ are $15°$ , and $5$ , respectively.

Explanation of Solution

Given:

The given figure is as follows.

  

Calculation:

From the figure it can be observed that $\angle A\cong \angle DBA$ , so use the Converse of the Base Angle theorem.

By the Converse of the Base Angle theorem,

If two angles of a triangle are congruent, then the sides opposite to them are congruent. So,

  $\begin{array}{c}\overline{BD}\cong \overline{AD}\\ \overline{BD}=26\end{array}$

From the figure it can be observed that, all the angles of upper triangle $\Delta BDC$ are congruent, so the triangle forms from an equilateral triangle. So, all the sides of the triangle are also congruent.

  $\begin{array}{c}BC=BD\\ 5y+1=26\\ 5y=25\\ y=5\end{array}$

Since $\Delta BDC$ is an equilateral triangle, so each angle is equal to $60°$ , so $\angle BDC=60°$ . So by Linear Pair Postulates,

  $\begin{array}{c}\angle ABD+\angle BDC=180°\\ 8x°+60°=180°\\ 8x°=180°-60°\\ x°=15°\end{array}$

Therefore, the values of $x$ and $y$ are $15°$ , and $5$ , respectively.