To determine which statement is TRUE, let's analyze the triangles and the given postulates.

Step 1: Identifying Triangle Similarity

Triangles can be similar by:

1. Side-Angle-Side (SAS): If two sides are proportional and the included angle is congruent.
2. Angle-Angle (AA): If two angles of one triangle are congruent to two angles of another triangle.

From the diagram:

• The angles included between corresponding sides appear congruent.
• The sides are labeled with their lengths.

Step 2: Checking Side Ratios

Let’s compare the ratios of corresponding sides for the two triangles:

• Side 1: $5$ and $y$
• Side 2: $4$ and $x$
• Side 3: $6$ and $7$

From options B and C, the ratios given are:
$x = \frac{42}{5} \quad \text{and} \quad y = \frac{20}{7}$ We’ll confirm this ratio.

Step 3: Verifying Statements

The statements are as follows:

• A and D: Use Side-Angle-Side (SAS) and give values $x = \frac{30}{7}$ and $y = \frac{28}{5}$.
• B and C: Use Angle-Angle (AA) and give values $x = \frac{42}{5}$ and $y = \frac{20}{7}$.

Since angle-angle similarity is easiest to confirm (AA), and the ratios align with $x = \frac{42}{5}$ and $y = \frac{20}{7}$, statements B and C are likely correct.

However:

• B refers to the AA postulate, which matches the problem setup.

Correct Answer:

B. The triangles are similar based on the angle-angle similarity postulate; $x = \frac{42}{5}$ and $y = \frac{20}{7}$.

Let me know if you need further clarification!

Related Questions:

1. What are the conditions for triangles to be similar?
2. How does the AA postulate work in triangle similarity?
3. What does the SAS postulate require for proving similarity?
4. How do you calculate side lengths in similar triangles?
5. Why is the angle-angle (AA) postulate more straightforward to prove?

Tip: Always compare the ratios of corresponding sides carefully when determining triangle similarity.