Which of the following statements correctly describes the requirement for two triangles to be proven similar by the $SAS$ (Side-Angle-Side) similarity theorem?

• A. All three pairs of corresponding angles are equal.
• B. All three pairs of corresponding sides are in proportion.
• C. Two pairs of corresponding angles are equal and one pair of sides are in proportion.
• D. Two pairs of corresponding sides are in proportion and the included angles are equal.

Answer: (D)

Verified step by step guidance

1. Recall that the SAS (Side-Angle-Side) similarity theorem states that two triangles are similar if two pairs of corresponding sides are in proportion and the included angle between those sides is equal.
2. Identify the key components of the SAS similarity condition: it requires two sides and the angle between them, not just any sides or angles.
3. Understand that 'included angle' means the angle formed between the two sides being compared, which is crucial for the SAS similarity to hold.
4. Compare the given options to the SAS similarity criteria: the correct statement must mention two pairs of sides in proportion and the included angle equal.
5. Conclude that the correct description is: 'Two pairs of corresponding sides are in proportion and the included angles are equal,' as this matches the SAS similarity theorem exactly.