Chapter 12.3, Problem 29E

To determine

To find:

Thevalues of $x$ and $y$ .

Answer to Problem 29E

  $x=4$

  $y=5$ .

Explanation of Solution

Given:

  $\begin{array}{l}\angle ACB\cong \angle DCE\\ \Delta ACB\cong \Delta DCE\end{array}$

Concept used:

Two geometric figures are congruent if a rigid motion or a composition of rigid motions maps one of the figures onto other.

Calculation:

The pairs of congruent corresponding angles

  $\begin{array}{l}\angle ACB\cong \angle DCE\\ \Delta ACB\cong \Delta DCE\end{array}$

The pairs of congruent corresponding sides

  $\begin{array}{l}\overline{AC}=\overline{DC}\\ \overline{BC}=\overline{EC}\end{array}$

  $\begin{array}{l}2x+6=4y-6\\ 4x=3y+1\end{array}$

Solving the system of equation

  $\begin{array}{l}2x+6=4y-6\\ 4y=2x+12\\ 2y=x+6\\ y=\frac{x+6}{2}\\ 4x=3\times \frac{x+6}{2}+1\\ 4x=\frac{3x+18+2}{2}\\ 8x=3x+20\\ 5x=20\\ x=4\end{array}$

Substracting $x$ in one of the two equation

  $\begin{array}{l}y=\frac{x+6}{2}\\ y=\frac{4+6}{2}\\ y=\frac{10}{2}\\ y=5\end{array}$

Therefore,

The values of $x=4$ and $y=5$ .

  $\Delta ACB\cong \Delta DCE$