Asked by Ian
Explain how the SAS criterion can be used to prove similarity in two triangles. A. If all three sides lengths are given and proportional to one another, resulting in the same factor, then the triangles are similar. B. If two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar. C. If at least two angles of the triangle are given and are the same, then the triangles are similar. D. If given a scale factor, you can determine if triangles are similar.
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Start with the SAS similarity statement and a short proof idea, then comment on each option.
SAS similarity criterion
- If in triangles ABC and DEF you have AB/DE = AC/DF and ∠A = ∠D (the angle included between the given sides), then the triangles are similar.
- Proof idea (scaling): scale triangle DEF by the factor k = AB/DE about D so that DE is sent to a segment DE' = AB and DF is sent to DF' = k·DF = AC. Because the included angle at D is unchanged by scaling (∠D = ∠A), triangles ABC and DE'F' are congruent by SAS, hence all corresponding angles match. Therefore the original triangles ABC and DEF are similar (their corresponding angles match and corresponding sides are proportional).
Comments on the options
A. True, but this is SSS similarity, not SAS. If all three side pairs are proportional by the same factor, the triangles are similar (SSS~).
B. True. This is exactly the SAS similarity criterion: two sides proportional and the included angle equal implies similarity.
C. True. If two angles of one triangle equal two angles of another, the third angles are equal too, so the triangles are similar (AA similarity).
D. Ambiguous but essentially true if interpreted correctly. If you are given a uniform scale factor that maps each side of one triangle to the corresponding side of the other, then the triangles are similar. However, just “a scale factor” without specifying how it applies to corresponding sides/angles is incomplete information by itself.
Bottom line: B is the direct SAS criterion; A and C are other standard similarity tests; D is true only when the scale factor consistently relates corresponding sides.
SAS similarity criterion
- If in triangles ABC and DEF you have AB/DE = AC/DF and ∠A = ∠D (the angle included between the given sides), then the triangles are similar.
- Proof idea (scaling): scale triangle DEF by the factor k = AB/DE about D so that DE is sent to a segment DE' = AB and DF is sent to DF' = k·DF = AC. Because the included angle at D is unchanged by scaling (∠D = ∠A), triangles ABC and DE'F' are congruent by SAS, hence all corresponding angles match. Therefore the original triangles ABC and DEF are similar (their corresponding angles match and corresponding sides are proportional).
Comments on the options
A. True, but this is SSS similarity, not SAS. If all three side pairs are proportional by the same factor, the triangles are similar (SSS~).
B. True. This is exactly the SAS similarity criterion: two sides proportional and the included angle equal implies similarity.
C. True. If two angles of one triangle equal two angles of another, the third angles are equal too, so the triangles are similar (AA similarity).
D. Ambiguous but essentially true if interpreted correctly. If you are given a uniform scale factor that maps each side of one triangle to the corresponding side of the other, then the triangles are similar. However, just “a scale factor” without specifying how it applies to corresponding sides/angles is incomplete information by itself.
Bottom line: B is the direct SAS criterion; A and C are other standard similarity tests; D is true only when the scale factor consistently relates corresponding sides.
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