Question

decide whether the triangles are similar. if so, determine the appropriate expression to solve for x. the triangles are not similar; no expression for x can be found △abc - △def, x = r·w/u △abc - △efd, x = r·w/z △abc - △fde, x = z·w/r

Explanation:

Step1: Find angles in $\triangle ABC$

In $\triangle ABC$, given $\angle A = 53^{\circ}$ and $\angle C=62^{\circ}$. Using the angle - sum property of a triangle ($\angle A+\angle B+\angle C = 180^{\circ}$), we find $\angle B=180^{\circ}-(53^{\circ}+62^{\circ}) = 65^{\circ}$.

Step2: Match angles for similarity

In $\triangle ABC$, angles are $53^{\circ},62^{\circ},65^{\circ}$ and in $\triangle EFD$, angles are $53^{\circ},65^{\circ}$ (given). So, $\triangle ABC\sim\triangle EFD$ by the AA (angle - angle) similarity criterion.

Step3: Set up proportion for similar sides

For similar triangles $\triangle ABC$ and $\triangle EFD$, the ratios of corresponding sides are equal. If $\frac{AB}{EF}=\frac{BC}{FD}=\frac{AC}{ED}$, we have $\frac{w}{z}=\frac{x}{r}$. Cross - multiplying gives $x = r\cdot\frac{w}{z}$.

Answer:

$\triangle ABC\sim\triangle EFD,x = r\cdot\frac{w}{z}$