Question

if \\( \triangle jkl \\) is isosceles, what is \\( m \angle jkl \\)?

Explanation:

Step1: Set up equation for base angles

Since $\triangle JKL$ is isosceles with two equal - length sides (marked), $\angle J=\angle L$. So, $10x - 13=3x + 22$.

Step2: Solve for $x$

Subtract $3x$ from both sides: $10x-3x - 13=3x-3x + 22$, which gives $7x-13 = 22$.
Add $13$ to both sides: $7x-13 + 13=22 + 13$, so $7x=35$.
Divide both sides by $7$: $x=\frac{35}{7}=5$.

Step3: Find $\angle J$ and $\angle L$

Substitute $x = 5$ into $\angle J$: $\angle J=10x-13=10\times5 - 13=37^{\circ}$.
Since $\angle J=\angle L = 37^{\circ}$.

Step4: Use triangle - angle sum theorem

The sum of angles in a triangle is $180^{\circ}$. Let $\angle JKL=y$. Then $y+\angle J+\angle L=180^{\circ}$.
Substitute $\angle J = 37^{\circ}$ and $\angle L = 37^{\circ}$: $y+37^{\circ}+37^{\circ}=180^{\circ}$.
Simplify: $y=180-(37 + 37)=106^{\circ}$.

Answer:

$106^{\circ}$