Questions: Similar Triangles: SSS and SAS Similarity

Question 1

Triangle ABC has sides 6, 9, and 12. Triangle DEF has sides 4, 6, and 8. Which statement best describes the relationship between these triangles?

• A. They are congruent by SSS because all three side pairs are given
• B. They are similar by SSS because all three side ratios are equal (3/2)
• C. They cannot be compared without knowing the angles
• D. They are similar, but the criterion is AA, not SSS

Answer: (B)

All three side ratios are equal: 6/4 = 9/6 = 12/8 = 3/2. This satisfies SSS Similarity. Note that option A confuses similarity with congruence — congruence requires equal sides, not proportional ones. Since the sides are proportional but not equal, the triangles are similar (same shape, different size), not congruent. No angle information is needed when all three side ratios are confirmed equal.

Question 2

Triangle PQR has PQ = 5, PR = 10, and angle P = 40°. Triangle XYZ has XY = 3, XZ = 6, and angle Y = 40°. Are these triangles similar by SAS?

• A. Yes — the side ratios are equal and a congruent angle exists in each triangle, which is sufficient for SAS similarity
• B. No — the 40° angle in triangle XYZ is angle Y, not the included angle between XY and XZ (which would be angle X)
• C. Yes — two side ratios are equal (5/3 = 10/6) and a 40° angle appears in each triangle
• D. No — SAS similarity requires all three side ratios to be equal, not just two

Answer: (B)

The side ratios are equal (5/3 = 10/6), but SAS similarity requires the congruent angle to be the INCLUDED angle — sandwiched between the two proportional sides. In triangle PQR, angle P is between PQ and PR: it is the included angle. In triangle XYZ, the included angle between XY and XZ would be angle X, not angle Y. Since the 40° angle in XYZ is at vertex Y, not vertex X, the SAS criterion is not satisfied. This is the most common SAS error.

Question 3

If all three pairs of corresponding sides of two triangles are in the same ratio, the triangles must be similar even if no angle measures are given.

• T. True
• F. False

Answer: True

This is exactly what SSS Similarity states: proportional corresponding sides alone are sufficient to establish similarity — no angle information is needed. Compare this with SSS Congruence, which requires equal (not merely proportional) corresponding sides. SSS Similarity is a complete, self-sufficient criterion.

Question 4

In SAS Similarity, any pair of congruent angles combined with two proportional side pairs is sufficient, regardless of which angle it is.

• T. True
• F. False

Answer: False

SAS Similarity specifically requires the congruent angle to be the INCLUDED angle — the one formed between the two proportional sides. If the congruent angle is not included (not between those two sides), the triangles may not be similar. This mirrors how SSA fails as a congruence criterion: the position of the angle relative to the sides determines whether the criterion holds. Always verify that the congruent angle is sandwiched between the two proportional side pairs.

Question 5

Explain the key difference between SSS Congruence and SSS Similarity, and describe how you would set up the check for SSS Similarity given two triangles.

Model answer: SSS Congruence requires all three pairs of corresponding sides to be equal in length; SSS Similarity requires all three pairs to be in the same ratio. To check SSS Similarity, pair corresponding sides (smallest-to-smallest, largest-to-largest, or by vertex labeling), compute the three ratios, and verify they are all equal. If they are, the triangles are similar with that ratio as the scale factor.

The shift from 'equal' to 'proportional' is the heart of similarity. Congruence is similarity with scale factor 1. Students who confuse the two often try to use equal side lengths to establish similarity, or mistakenly test for congruence when similarity is called for. Setting up ratios consistently — always putting the same triangle's sides in the numerator — prevents the arithmetic errors that come from mixing the orientation.