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SSS and SAS Similarity: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

No vertex matching is supplied. Triangles have side lengths $5, 8, 10$ and $7.5, 12, 16$ . Which conclusion is correct?

Answer choices for question 1 A They are similar because two ratios equal 1.5 B They are not similar under any matching C The largest sides may be ignored D They must be congruent
2

For one proposed SAS correspondence, the target-to-original side ratios are $5/3$ and $3/5$ , and the included angles are equal. What fails?

Answer choices for question 2 A The included angles have to be unequal B The common factor is one C The two side ratios are unequal D An equal angle is never useful for similarity
3

Under a specified correspondence, original sides are $x+2$ , $x+6$ , and $10$ , while image sides are $9$ , $15$ , and $y$ . Which pair makes SSS similarity hold with positive triangle lengths?

Answer choices for question 3 A $x=4$ and $y=15$ B $x=4$ and $y=10$ C $x=6$ and $y=15$ D $x=2$ and $y=20$
4

All three corresponding side ratios equal 2. A given equal angle pair happens to be nonincluded for two selected sides. Which conclusion is justified?

Answer choices for question 4 A No similarity criterion can apply B The triangles must be dissimilar because the angle is nonincluded C SAS is the only possible criterion D SSS similarity already applies; the nonincluded angle is not needed
5

In an actual counterexample, two side ratios equal 1 and a nonincluded angle pair is equal, but the complete angle sets differ. What does this show?

Answer choices for question 5 A That nonincluded-angle information is not a universal similarity criterion B No pair with nonincluded-angle information can ever be similar C SAS with an included angle is invalid D All triangles with factor 1 are dissimilar
6

Two right triangles have legs $5, 12$ and $8, 19.2$ . Which criterion establishes similarity without finding either hypotenuse?

Answer choices for question 6 A SSS, because two side pairs count as three B SAS, using the equal right angles between the legs C No criterion until the hypotenuses are calculated D A nonincluded-angle rule
7

Triangles satisfy
$\begin{aligned}AB&=AC\\DE&=DF\\\angle B&=\angle E,\\&=52^\circ.\end{aligned}$
Why can similarity be proved despite the given angle being nonincluded for those equal-side pairs?

Answer choices for question 7 A Any nonincluded angle always gives SAS B The triangles must have equal side lengths across both figures C An unknown angle can be assumed equal D The isosceles base-angle results give a second angle pair, so AA applies
8

Triangles have equal included angles at $A,D$ , with
$\begin{aligned}AB&=x+1\\AC&=2x-1\\DE&=12\\DF&=18.\end{aligned}$
Which positive value of $x$ makes the two SAS ratios agree?

Answer choices for question 8 A $2$ B $3$ C $5$ D $7$
9

Why is zero excluded from the common factor in SSS and SAS similarity?

Answer choices for question 9 A Zero reverses every triangle B Scaling by zero collapses every triangle to one point C Zero gives a rigid rotation D Zero preserves three positive side lengths
10

A proposed SSS proof starts by assuming that proportional sides already imply equal angles, then uses the claimed similarity to justify those angles. Which repair avoids that circular step?

Answer choices for question 10 A Dilate to match the three lengths and use earlier SSS congruence B Assume the conclusion twice C Measure the angles from a sketch D Omit the third side ratio
11

It is given that both $AB/DE$ and $AC/DF$ equal $2/5$ , and $\angle A\cong\angle D$ . What is the factor from $ABC$ to $DEF$ under SAS?

Answer choices for question 11 A $\frac25$ B $\frac35$ C $\frac52$ D $\frac72$
12

One triangle has side lengths $5, 7, 11$ . Another has side lengths $10, 22,q$ , with no correspondence specified. Which $q$ makes the triangles similar?

Answer choices for question 12 A $15$ B $17$ C $24$ D $14$

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