Math 2 Quiz: Triangle Similarity Criteria

What this quiz covers

This quiz focuses on Triangle Similarity Criteria, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The third angle of Triangle 2, which must be $73°$ for the triangles to be similar
2. At least one pair of corresponding side lengths to confirm the angle correspondence (correct answer)
3. All three side lengths of both triangles to verify proportionality ratios
4. No additional information needed, since two angles match and AA similarity requires only two pairs

Question 2

1. SSS similarity, because the ratios of corresponding sides are $\frac{3}{5}$ for all three pairs (correct answer)
2. SSS similarity, because the ratios of corresponding sides are $\frac{4}{5}$ for all three pairs
3. SAS similarity, because two pairs of sides are proportional and included angles are equal
4. No similarity criterion applies because the triangles have different side length ratios

Question 3

1. Only triangle $JKL$ is similar, because its sides are in ratio $3:4:5$ (correct answer)
2. Only triangle $MNP$ is similar, because its sides are in ratio $3:5:6$
3. Both triangles $JKL$ and $MNP$ are similar to triangle $GHI$
4. Neither triangle is similar to triangle $GHI$ because the ratios don't match exactly

Question 4

1. The student made an error calculating the hypotenuse of Triangle 2, which should be 36, not 39
2. The student made an error in the ratio calculation; the actual ratio is $\frac{1}{2}$, not $\frac{1}{3}$
3. The student made an error by using SSS; these triangles should be proven similar using SAS instead
4. The student made no error; the hypotenuses are correct and all side ratios equal $\frac{1}{3}$, confirming similarity (correct answer)

Question 5

1. The ratios are actually $\frac{5}{3}$, $\frac{3}{2}$, and $\frac{5}{3}$, so they are not all equal (correct answer)
2. The ratios are actually $\frac{5}{3}$, $\frac{2}{3}$, and $\frac{3}{5}$, so SSS similarity does not apply
3. The student incorrectly ordered the sides; the correct ratios are $\frac{4}{3}$, $\frac{3}{2}$, and $\frac{5}{3}$
4. The calculation is correct but the triangles are not similar because SSS requires equal ratios of less than 2

Question 6

1. The conclusion is definitely correct because both triangles have the same three angle measures
2. The conclusion is correct, but the student should specify which angles correspond to avoid confusion (correct answer)
3. The conclusion is incorrect because isosceles triangles with the same angles are not necessarily similar
4. The conclusion is correct by AA, and the triangles are also congruent since all angles match exactly

Question 7

1. The claim is correct because two pairs of corresponding angles are equal
2. The claim is incorrect because the third angles are not equal, so AA similarity does not apply
3. The claim is incorrect because we need three pairs of equal angles for similarity
4. The claim is correct because angle $C = 67°$ and angle $E = 48°$, giving two pairs of equal angles (correct answer)

Question 8

1. The claim is correct by SAS, and we can verify by calculating the hypotenuses: $RT = 50$ and $UW = 30$
2. The claim is incorrect because we need the hypotenuse lengths to confirm the third side ratio
3. The claim is correct because $\frac{RS}{UV} = \frac{ST}{VW} = \frac{5}{3}$ and both have right angles at corresponding vertices (correct answer)
4. The claim is incorrect because the right angles are not necessarily between the same corresponding sides

Question 9

1. All three triangles are similar because they all have the same shape ratios
2. Triangles $ABC$ and $DEF$ are similar with ratio $\frac{4}{3}$, but triangle $GHI$ is not similar to either
3. Triangles $ABC$ and $GHI$ are similar with ratio $\frac{4}{5}$, but triangle $DEF$ is not similar to either
4. All three triangles are similar: $ABC \sim DEF$ with ratio $\frac{4}{3}$ and $ABC \sim GHI$ with ratio $\frac{4}{5}$ (correct answer)

Question 10

1. The student failed to verify that all three angle pairs are equal
2. The student failed to establish proper correspondence between the triangles (correct answer)
3. The student should have used SAS similarity instead of AA similarity
4. The student failed to calculate the third angle in Triangle 2

Question 11

1. No error; the triangles are similar by SAS with scale factor $\frac{3}{5}$ (correct answer)
2. The student used the wrong similarity criterion; AA should be used instead
3. The student incorrectly assumed angle correspondence without verification
4. The side ratios are not equal, so SAS similarity does not apply

Question 12

1. $JL = 15$
2. $JL = 20$
3. $JL = 30$ (correct answer)
4. $JL = 36$

Question 13

1. Side $BC$ must be proportional to side $EF$ with the same ratio as the other sides
2. Angle $B$ must equal angle $E$
3. Angle $C$ must equal angle $F$
4. The given conditions are sufficient; no additional condition is needed (correct answer)

Question 14

1. They are similar, but the similarity criterion cannot be determined from the given information
2. They are similar by SAS because the 90° angle corresponds to the right angle in triangle XYZ
3. They are not similar because insufficient information is given about triangle MNP
4. They are similar by SSS because triangle MNP completes the 3:4:5 ratio (correct answer)

Question 15

1. Correct reasoning and correct scale factor of $\frac{2}{3}$
2. Correct reasoning but incorrect scale factor; should be $\frac{3}{4}$
3. Correct reasoning but incorrect scale factor; should be $\frac{4}{3}$ (correct answer)
4. Incorrect reasoning because the ratios are not proportional

Question 16

1. SAS similarity applies because $\frac{XY}{RS} = \frac{YZ}{ST} = \frac{3}{4}$ and $\angle Y = \angle S$
2. SAS similarity applies because $\frac{XY}{RS} = \frac{YZ}{ST} = \frac{4}{3}$ and $\angle Y = \angle S$
3. SAS similarity does not apply because we cannot determine if the angles are between corresponding proportional sides (correct answer)
4. SAS similarity applies because the ratios $\frac{21}{15} = \frac{28}{20} = 1.4$ and angles are equal