Question

Find the values of x and y.

29. In $\mathit{\Delta }\mathbf{JKL}\mathbf{,}\mathbf{}\overline{\mathrm{JK}}\cong \overline{\mathrm{KL}},\mathbf{m}\mathbf{\angle }\mathbf{J}\mathbf{=}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{,}\mathbf{}\mathbf{m}\mathbf{\angle }\mathbf{K}\mathbf{=}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{y}$and$\mathbf{m}\angle \mathbf{L}\mathbf{=}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{y}$

Short answer

The value of$x$ and$y$ is 30and 10 respectively.

Step-by-step solution

– Apply isosceles triangle theorem.

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

In $\Delta JKL,\overline{JK}\cong \overline{KL}$implies that $m\angle J=m\angle L$.

– Description of step.

Substitute$2x-y$ for$m\angle J$ and$x+2y$ for$m\angle L$ into$m\angle J=m\angle L$ and simplify.

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

– Apply an angle sum theorem.

The sum of measures of all interior angles of a triangle is 180.

Therefore,

$\begin{array}{c}\mathrm{m}\angle \mathrm{J}+\mathrm{m}\angle \mathrm{K}+\mathrm{m}\angle \mathrm{L}=180\\ 2\mathrm{x}-\mathrm{y}+2\mathrm{x}+2\mathrm{y}+\mathrm{x}+2\mathrm{y}=180\\ 5\mathrm{x}+3\mathrm{y}=180\end{array}$

– Description of step.

Substitute$3y$ for$x$ into$5x+3y=180$ to find the value of role="math" localid="1648815869555" $x$.

$\begin{array}{c}5\left(3y\right)+3y=180\\ 15y+3y=180\\ 18y=180\\ y=10\end{array}$

– Description of step.

Substitute 10 for$y$ into $x=3y$, to find the value of $x$.

$\begin{array}{c}x=3\times 10\\ =30\end{array}$

Therefore, the value of$x$ and$y$ is 30 and 10 respectively.