To prove that two triangles are similar by the SAS similarity theorem, it needs to be shown that two pairs of corresponding sides are in proportion and the included angles are equal. Which of the following statements correctly expresses this condition?

• A. $\frac{a}{a\text{'}}=\frac{b}{b\text{'}} \text{and} \angle C=\angle C\text{'}$
• B. $\frac{a}{A}=\frac{b}{B} \text{and} C=C$
• C. $a=a\text{'} \text{and} b=b\text{'}$
• D. $\frac{a}{b}=\frac{A}{B} \text{and} \angle A=\angle B$

Answer: (A)

Verified step by step guidance

1. Recall the SAS (Side-Angle-Side) similarity theorem: Two triangles are similar if two pairs of corresponding sides are in proportion and the included angles between those sides are equal.
2. Identify the corresponding sides in the two triangles, say sides \(a\) and \(b\) in the first triangle, and their counterparts \(a'\) and \(b'\) in the second triangle.
3. Express the proportionality condition as the equality of ratios of corresponding sides: \(\frac{a}{a'} = \frac{b}{b'}\).
4. Identify the included angle between these two sides in each triangle, say angle \(C\) in the first triangle and angle \(C'\) in the second triangle, and state that these angles must be equal: \(\angle C = \angle C'\).
5. Combine these conditions to correctly express the SAS similarity condition: \(\frac{a}{a'} = \frac{b}{b'}\) and \(\angle C = \angle C'\).