https://file.aiquickdraw.com/mathbot/file/1726758358099m9ldsxwt.jpg

To solve this question about the similarity of triangles \( \triangle PQR \) and \( \triangle STU \), we need to analyze the given information:

- \( \angle P \cong \angle S \) (one pair of corresponding angles are congruent).
- Side lengths:
  - \( PR = 12 \), \( SU = 3 \),
  - \( PQ = 20 \), \( ST = 5 \).

### Step-by-step approach:

1. **Check for a pair of congruent angles**:
   - Since \( \angle P \cong \angle S \), there is at least one pair of corresponding congruent angles.

2. **Compare the side ratios**:
   - The sides \( PR \) and \( SU \) correspond, so we calculate their ratio:
     \[
     \frac{PR}{SU} = \frac{12}{3} = 4.
     \]
   - Similarly, for the sides \( PQ \) and \( ST \):
     \[
     \frac{PQ}{ST} = \frac{20}{5} = 4.
     \]
   - The ratios of corresponding sides are equal, indicating proportionality.

3. **Determine the similarity postulate**:
   - Since one pair of corresponding angles is congruent, and the sides around that angle are proportional, we can apply the **SAS similarity theorem** (Side-Angle-Side similarity).

Thus, the triangles \( \triangle PQR \) and \( \triangle STU \) are **similar** by the **SAS similarity theorem**.

### Final Answer:
The correct option is:
- **B. Similar - SAS**

Would you like further explanation or details?

Here are 5 related questions:
1. How do you determine if two triangles are similar using the AA similarity postulate?
2. Can triangles be similar if only two sides are proportional?
3. What is the difference between similarity and congruence in triangles?
4. How does the SSS similarity postulate differ from the SAS postulate?
5. When is the AA postulate sufficient for triangle similarity?

**Tip:** Always check both angles and side ratios carefully to apply the correct similarity theorem.

David drew △PQR and △STU so that ∠P ≅ ∠S, PR = 12, SU = 3, PQ = 20, and ST = 5. Are △PQR and △STU similar? If so, identify the similarity postulate or theorem that applies.

This problem tests understanding of triangle similarity using the SAS similarity theorem. Given ∠P ≅ ∠S and corresponding side lengths of △PQR and △STU, solve for their similarity using the ratio of sides and congruent angles. Suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation