12 multiple-choice questions, progressively harder.
What length evidence is sufficient for SSS similarity of two nondegenerate triangles?
Solution
Correct answer: C
SSS similarity uses one common positive factor for all three matching side pairs. Two side pairs alone are not enough.
Which description states SAS similarity correctly?
Correct answer: A
The equal angle must lie between the two sides being compared in each triangle. The common side ratio is positive.
Which is the included angle for sides ABABAB and ACACAC?
Correct answer: B
The two selected sides share endpoint A. The included angle is therefore angle BAC, at A.
The matching side lengths are 7,9,127, 9, 127,9,12 in one triangle and 14,18,2414, 18, 2414,18,24 in the other. What is the scale factor from the first to the second?
Correct answer: D
Every second-triangle length is twice its matching first-triangle length.
What distinguishes SSS congruence from SSS similarity?
SSS congruence gives a length factor of one. SSS similarity allows any common positive factor, including one.
Only two pairs of proportional sides are known in two triangles. No angle information is supplied. What is justified?
Two matching side ratios leave the included angle or third-side ratio unconstrained. Similarity may hold, but it has not been established.
For a proposed correspondence, the three side ratios are 1.251.251.25, 1.251.251.25, and 1.41.41.4. Does SSS justify that correspondence?
SSS needs one common ratio across all three corresponding side pairs. The third ratio differs, so this proposed correspondence fails the test.
Two side ratios agree and a pair of nonincluded angles is equal. Which statement is always valid?
The equal angle is not in the required SAS position. Some such pairs are similar and some are not, so the evidence is not a general guarantee.
In the SSS similarity proof, what justifies a rigid-motion match after the dilation has made all three side pairs equal?
The earlier SSS congruence criterion applies to the resized triangle and the target. Congruence supplies the rigid motions needed to complete the similarity mapping.
In the SAS similarity proof, two side pairs have become equal after dilation and their included angles remain equal. Which earlier result applies?
Two equal side pairs with their equal included angles give SAS congruence. This is the prior result used to prove the new similarity criterion.
SSS has established a similarity in which an original side of length 999 matches an image side of length 151515. What is the scale factor from the original to the image?
Put image length over original length.
k=159,=53.\begin{aligned}k&=\frac{15}{9},\\&=\frac 53.\end{aligned}k=915,=35.
Which side-length triple could belong to a nondegenerate triangle in an SSS similarity problem?
The two shortest positive sides must sum to more than the longest.
The other triples are flat or cannot close, so the nondegenerate-triangle condition fails.
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