Test Question:

The triangles $\triangle PQR$ and $\triangle XYZ$ are shown in the image. Both triangles have their angles marked. Use the information given to answer the following:

1. Are the triangles $\triangle PQR$ and $\triangle XYZ$ similar? If so, explain why using a similarity criterion (e.g., AA, SSS, or SAS).

2. If $\triangle PQR \sim \triangle XYZ$, write the corresponding pairs of angles and sides that match.

3. Assume the side lengths of $\triangle PQR$ are $PQ = 5$, $QR = 6$, and $PR = 7$. If $\triangle XYZ$ is similar to $\triangle PQR$ and the side $XY = 10$, find the lengths of $YZ$ and $XZ$.

4. Prove why the sum of the angles in each triangle equals 180°, using the given diagram.

5. Determine whether the orientation of the two triangles affects their similarity.

Would you like me to solve or elaborate on any of these?

Five Related Questions:

1. How can you use proportions to solve for missing sides in similar triangles?
2. What does the AA criterion for triangle similarity state?
3. If two triangles are congruent, how do their side lengths compare?
4. Can triangles be similar if only two sides are proportional? Why or why not?
5. How does the concept of corresponding angles apply in other geometric figures?

Tip:

In similar triangles, the ratios of corresponding side lengths are always equal!