Triangle ABC is similar to triangle A'B'C'. Select all statements that must be true.

Steps to Solve

1. Understanding Triangle Similarity

Triangles ABC and A'B'C' are similar, which means all corresponding angles are equal and the ratios of the lengths of corresponding sides are proportional.

2. Identifying Corresponding Sides for Ratios

For similar triangles, we can establish the following relationships for corresponding sides:

• $ \frac{AB}{A'B'} = \frac{BC}{B'C'} = \frac{CA}{C'A'} $

3. Analyzing Each Statement

We will analyze each statement based on the properties of similar triangles:

• Statement A: $ \frac{BC}{B'C'} = \frac{AC}{A'C'} $

This statement is true as it directly represents one of the characteristics of similar triangles.

• Statement B: $ \frac{AB}{A'B'} = \frac{CA}{C'A'} $

This statement is also true by the similarity property.

• Statement C: $ \frac{B'C'}{BA} = \frac{B'A'}{BC} $

This statement rearranges the ratios and is not generally valid; it's not necessarily a true statement.

• Statement D: $ \frac{AB}{B'C'} = \frac{BC}{A'B'} $

This statement compares sides in a way that doesn't conform to the similarity rules; hence, it's false.

• Statement E: $ \frac{AC}{BC} = \frac{A'C'}{B'C'} $

This statement rearranges ratios that do not align with the property of similar triangles; thus, it's false.

4. Conclusion of Statements

Based on the analysis:

• A and B are true.
• C, D, and E are false.