Question

1. write a similarity statement for the triangles. then find the value of x.

Explanation:

Step1: Identify similar triangles

Since $\angle H=\angle K$ and the angles at $J$ are vertical - angles and thus equal, $\triangle GHJ\sim\triangle KLJ$ by the AA (Angle - Angle) similarity criterion.

Step2: Set up proportion

For similar triangles $\triangle GHJ$ and $\triangle KLJ$, the ratios of corresponding sides are equal. So, $\frac{GH}{KL}=\frac{GJ}{JL}$.
We know that $GH = 6$, $KL = 4$, $GJ = 9$, and $JL=x$. Substituting these values into the proportion gives $\frac{6}{4}=\frac{9}{x}$.

Step3: Cross - multiply

Cross - multiplying the proportion $\frac{6}{4}=\frac{9}{x}$ gives $6x=4\times9$.

Step4: Solve for x

First, calculate $4\times9 = 36$. Then, we have the equation $6x = 36$. Divide both sides of the equation by 6: $x=\frac{36}{6}=6$.

Answer:

The similarity statement is $\triangle GHJ\sim\triangle KLJ$ and $x = 6$.