Let's go through the questions step-by-step based on what I can see in the image.

Analysis:

1. Question 9: This question is about determining if two triangles are similar and then selecting the correct similarity statement and similarity ratio.

   • Given: In triangles ∆QRS and ∆UVT, QR = 10, RS = 15, and ∠R = 32°. In triangle ∆UVT, UV = 12, VT = 18, and ∠U = 32°.

   • Solution: To determine similarity, we can use the SAS (Side-Angle-Side) similarity postulate if the included angle and proportional sides match.

   • Similarity check: $\frac{QR}{UV} = \frac{10}{12} = \frac{5}{6} \quad \text{and} \quad \frac{RS}{VT} = \frac{15}{18} = \frac{5}{6}$ Since both pairs of sides are proportional and the included angle ∠R matches ∠U (both 32°), the triangles are similar by SAS similarity.

   • Similarity statement and ratio: The correct option would likely be (c) ∆QRS ~ ∆UVT, 5/6.

2. Question 10:

   • Problem: We have two triangles, ABC and DEF, which are similar, and we are asked to find the lengths of AB and EF.
   • Solution: Using the similarity ratio between corresponding sides, we can solve for the unknown values AB and EF using the provided similarity ratio and the measurements.

3. Question 11:

   • Problem: This question is about calculating the length of the shorter side of a golden rectangle given the longer side. The canvas is 33 inches.
   • Solution: The golden ratio is approximately 1.618. So, for a golden rectangle: $\text{Shorter Side} = \frac{33}{1.618} \approx 20.4 \text{ inches (rounded to the nearest tenth)}$

4. Question 12:

   • Problem: This question asks for the second dimension of a golden rectangle given the first measurement as 18 inches.
   • Solution: Using the golden ratio again, the possible second measurement can be calculated, and the correct answer should align with one of the options.

5. Question 13:

   • This question involves reasoning with similarity postulates to establish whether triangles are similar and which postulate was used.

Would you like me to solve each question in more detail, or do you have a specific question you’d like me to focus on?

Follow-up Questions

1. How does the SAS similarity postulate work for triangles?
2. Can you explain the golden ratio in relation to rectangles and proportions?
3. How can we calculate unknown sides in similar triangles using proportionality?
4. What is the importance of rounding to the nearest tenth in measurements?
5. How does the AA (Angle-Angle) postulate compare to SAS in determining triangle similarity?

Tip

When dealing with similar triangles, always look for proportional pairs of sides and matching angles as per the similarity criteria.