Triangle Similarity Criteria - Math 2

1. Triangle $JKL$ is similar to triangle $PQR$ with $J$ corresponding to $P$, $K$ corresponding to $Q$, and $L$ corresponding to $R$. If $JK = 18$, $KL = 24$, $PQ = 12$, and $QR = 16$, what is the length of $JL$?
   Answer: $JL = 30$ Since the triangles are similar with the given correspondence, all corresponding sides are proportional. We have JK/PQ = 18/12 = 3/2 and KL/QR = 24/16 = 3/2, confirming the scale factor is 3/2. To find the third side of triangle PQR, we use the fact that the sides are proportional: if JK:KL = 18:24 = 3:4, then PQ:QR:PR = 3:4:5 (completing the proportion). So PR = 20. Therefore, JL/PR = 3/2, which gives us JL = (3/2) × 20 = 30.
2. Triangle $XYZ$ has sides in the ratio $3:4:5$. Triangle $MNP$ has two sides of length 12 and 16, with the angle between them measuring 90°. What can be concluded about the relationship between these triangles?
   Answer: They are similar by SSS because triangle MNP completes the 3:4:5 ratio When you encounter triangles with given side ratios or specific angle measurements, think about triangle similarity and the Pythagorean theorem. These problems often test whether you can recognize special right triangles and apply similarity criteria. Triangle XYZ has sides in the ratio 3:4:5. This is the most famous Pythagorean triple, so XYZ is a right triangle. For triangle MNP, you have two sides (12 and 16) with a 90° angle between them. Using the Pythagorean theorem to find the third side: $c^2 = 12^2 + 16^2 = 144 + 256 = 400$, so $c = 20$. Triangle MNP has sides 12, 16, and 20. To check if these triangles are similar, compare the ratios: $\frac{12}{3} = 4$, $\frac{16}{4} = 4$, and $\frac{20}{5} = 4$. Since all ratios equal 4, the triangles have proportional sides and are similar by SSS (Side-Side-Side similarity). Choice A is wrong because we can determine the similarity criterion—it's SSS. Choice B incorrectly identifies SAS as the criterion; while both triangles have right angles, we're using the fact that all three pairs of sides are proportional. Choice C is wrong because we have sufficient information once we calculate the third side of triangle MNP using the Pythagorean theorem. Remember that when you see a 90° angle between two given sides, you can always find the third side using the Pythagorean theorem. Also, memorize common Pythagorean triples like 3-4-5 and their multiples—they appear frequently in geometry problems.
3. Two right triangles have the following measurements: Triangle 1 has legs of length 5 and 12. Triangle 2 has legs of length 15 and 36. A student calculates that the hypotenuses are 13 and 39 respectively, then claims the triangles are similar by SSS with ratio $\frac{1}{3}$. What error, if any, did the student make?
   Answer: The student made no error; the hypotenuses are correct and all side ratios equal $\frac{1}{3}$, confirming similarity First, verify the hypotenuses: Triangle 1: $\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ ✓. Triangle 2: $\sqrt{15^2 + 36^2} = \sqrt{225 + 1296} = \sqrt{1521} = 39$ ✓. Now check ratios: $\frac{5}{15} = \frac{1}{3}$, $\frac{12}{36} = \frac{1}{3}$, $\frac{13}{39} = \frac{1}{3}$. All ratios equal $\frac{1}{3}$, so the triangles are similar by SSS. The student's work and conclusion were entirely correct.
4. In triangle $XYZ$, the angles are $50°$, $80°$, and $50°$. Triangle $PQR$ has angles $50°$, $50°$, and $80°$. A student concludes these triangles are similar by AA. What should be considered about this conclusion?
   Answer: The conclusion is correct, but the student should specify which angles correspond to avoid confusion Both triangles have angles measuring $50°$, $50°$, and $80°$, so they are indeed similar by AA (actually AAA). However, good mathematical practice requires specifying the correspondence. For instance, if $\angle X = \angle P = 50°$ and $\angle Y = \angle Q = 80°$, then $\angle Z = \angle R = 50°$. Choice A is mathematically correct but lacks precision. Choice C is wrong—triangles with identical angle measures are always similar. Choice D incorrectly claims congruence (angles alone don't determine congruence).
5. Triangle $PQR$ has sides $PQ = 15$, $QR = 20$, and $PR = 25$. Triangle $STU$ has sides $ST = 9$, $TU = 12$, and $SU = 15$. Which similarity criterion, if any, can be used to prove these triangles are similar?
   Answer: SSS similarity, because the ratios of corresponding sides are $\frac{3}{5}$ for all three pairs Check the ratios: $\frac{ST}{PQ} = \frac{9}{15} = \frac{3}{5}$, $\frac{TU}{QR} = \frac{12}{20} = \frac{3}{5}$, and $\frac{SU}{PR} = \frac{15}{25} = \frac{3}{5}$. All three ratios are equal, so SSS similarity applies. Choice B uses the wrong ratio. Choice C is incorrect because we don't know any angles. Choice D is wrong because the ratios are actually equal.
6. Triangle $GHI$ has sides in the ratio $3:4:5$. Triangle $JKL$ has sides $JK = 15$, $KL = 20$, and $JL = 25$. Triangle $MNP$ has sides $MN = 9$, $NP = 15$, and $MP = 18$. Which triangles, if any, are similar to triangle $GHI$?
   Answer: Only triangle $JKL$ is similar, because its sides are in ratio $3:4:5$ Triangle $GHI$ has sides in ratio $3:4:5$. For triangle $JKL$: the ratio is $15:20:25 = 3:4:5$ (dividing by 5). For triangle $MNP$: the ratio is $9:15:18 = 3:5:6$ (dividing by 3). Only triangle $JKL$ has the same ratio as triangle $GHI$, so only it is similar by SSS. Choice B incorrectly identifies the ratio for $MNP$ and wrongly concludes it's similar. Choice C incorrectly includes both triangles. Choice D is wrong because $JKL$ does match the ratio.
7. Two triangles have the following measurements: Triangle 1 has angles of 45°, 60°, and 75°. Triangle 2 has two angles measuring 45° and 75°. A student immediately concludes the triangles are similar by AA. What is the most significant issue with this reasoning?
   Answer: The student failed to establish proper correspondence between the triangles While Triangle 2's third angle is 180° - 45° - 75° = 60°, making both triangles have angles 45°, 60°, and 75°, the critical issue is that the student didn't establish which angles correspond to which. There are multiple ways to match the angles, and without proper correspondence, we cannot determine the similarity transformation or scale factor. AA requires not just equal angles, but correctly identified corresponding angles.
8. Triangle $ABC$ has $AB = 15$, $BC = 20$, and angle $B = 53°$. Triangle $DEF$ has $DE = 9$, $EF = 12$, and angle $E = 53°$. A student claims these triangles are similar by SAS. What error, if any, did the student make?
   Answer: No error; the triangles are similar by SAS with scale factor $\frac{3}{5}$ For SAS similarity, we need two pairs of sides to be proportional with the included angles equal. We have AB/DE = 15/9 = 5/3 and BC/EF = 20/12 = 5/3. The ratios are equal, and angle B = angle E = 53° are the included angles between these sides. Therefore, SAS similarity applies with scale factor 5/3 (or 3/5 from ABC to DEF). The student's reasoning is correct.
9. Triangle $PQR$ has sides of length 6, 9, and 12. Triangle $STU$ has sides of length 8, 12, and 16. A student applies SSS similarity and concludes the triangles are similar with a scale factor of $\frac{2}{3}$. Which statement best describes this conclusion?
   Answer: Correct reasoning but incorrect scale factor; should be $\frac{4}{3}$ For SSS similarity, all corresponding side ratios must be equal. Triangle PQR has sides 6, 9, 12. Triangle STU has sides 8, 12, 16. The ratios are 8/6 = 4/3, 12/9 = 4/3, and 16/12 = 4/3. Since all ratios equal 4/3, the triangles are similar by SSS, but the scale factor from PQR to STU is 4/3, not 2/3. The student confused the direction of the ratio.
10. In triangle $ABC$, side $AB = 21$ and side $AC = 28$. In triangle $DEF$, side $DE = 15$ and side $DF = 20$. If angle $A = 65°$ and angle $D = 65°$, what additional condition must be satisfied for the triangles to be similar by SAS?
    Answer: The given conditions are sufficient; no additional condition is needed When you encounter triangle similarity problems, remember that there are three main similarity criteria: SSS (all sides proportional), AA (two angles equal), and SAS (two sides proportional with the included angle equal). For SAS similarity, you need two pairs of proportional sides with the included angles being equal. Let's check what we have: In triangle $ABC$, sides $AB = 21$ and $AC = 28$ form angle $A = 65°$. In triangle $DEF$, sides $DE = 15$ and $DF = 20$ form angle $D = 65°$. First, verify the sides are proportional: $\frac{AB}{DE} = \frac{21}{15} = \frac{7}{5} = 1.4$ and $\frac{AC}{DF} = \frac{28}{20} = \frac{7}{5} = 1.4$. The ratios are equal, so the sides are proportional. Since the included angles are both $65°$, we have satisfied all SAS requirements. Choice A is incorrect because while the third sides will be proportional (this follows from SAS), it's not an additional requirement—it's a consequence. Choice B is wrong because equal angles $B$ and $E$ would suggest AA similarity, not SAS, and these aren't the included angles anyway. Choice C is incorrect for the same reason as B—angles $C$ and $F$ aren't part of the SAS criteria we're using. The answer is D because we already have everything needed for SAS similarity: proportional sides with equal included angles. Study tip: For SAS similarity, focus on the included angle—the angle between the two given sides. Once you have proportional sides and equal included angles, you're done.
11. Two triangles have the following measurements: Triangle 1 has angles of $42°$, $73°$, and $65°$. Triangle 2 has two angles measuring $42°$ and $65°$. A student concludes that AA similarity applies. What additional information would be needed to verify this conclusion?
    Answer: At least one pair of corresponding side lengths to confirm the angle correspondence While Triangle 2's third angle is indeed $73°$ (making all angles match), the key issue is determining which angles correspond. The $42°$ and $65°$ angles in Triangle 2 could correspond to different angles in Triangle 1, potentially making the triangles non-similar despite having the same angle measures. We need additional information (like side lengths or angle positions) to establish proper correspondence. Choice A misses the correspondence issue. Choice C requires more than necessary. Choice D incorrectly assumes correspondence is obvious.
12. Triangles $ABC$ and $DEF$ are given with $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = \frac{5}{3}$. A student claims this proves the triangles are similar by SSS. However, upon checking, it's found that $AB = 20$, $BC = 15$, $AC = 25$, $DE = 12$, $EF = 10$, and $DF = 15$. What is wrong with the student's reasoning?
    Answer: The ratios are actually $\frac{5}{3}$, $\frac{3}{2}$, and $\frac{5}{3}$, so they are not all equal Calculate the actual ratios: $\frac{AB}{DE} = \frac{20}{12} = \frac{5}{3}$, $\frac{BC}{EF} = \frac{15}{10} = \frac{3}{2}$, and $\frac{AC}{DF} = \frac{25}{15} = \frac{5}{3}$. The ratios are not all equal ($\frac{5}{3} \neq \frac{3}{2}$), so SSS similarity does not apply. The student's error was assuming the ratios were all $\frac{5}{3}$ without calculating them. Choice B has incorrect ratios. Choice C also has wrong calculations. Choice D misunderstands SSS similarity requirements.
13. In triangle $ABC$, angle $A = 65°$ and angle $B = 48°$. In triangle $DEF$, angle $D = 65°$ and angle $F = 67°$. A student claims these triangles are similar by AA similarity. What can be concluded about this claim?
    Answer: The claim is correct because angle $C = 67°$ and angle $E = 48°$, giving two pairs of equal angles First, find the third angles: In triangle $ABC$, angle $C = 180° - 65° - 48° = 67°$. In triangle $DEF$, angle $E = 180° - 65° - 67° = 48°$. Now we have: $\angle A = \angle D = 65°$ and $\angle C = \angle F = 67°$. Two pairs of corresponding angles are equal, so AA similarity applies. Choice A is wrong because it doesn't verify which angles correspond. Choice B incorrectly calculates the third angles. Choice C misunderstands AA similarity (only two pairs needed).
14. Triangle $ABC$ has sides $AB = 12$, $BC = 16$, and $CA = 20$. Triangle $DEF$ has sides $DE = 9$, $EF = 12$, and $FD = 15$. Triangle $GHI$ has sides $GH = 15$, $HI = 20$, and $IG = 25$. Which statement about the similarity relationships among these triangles is correct?
    Answer: All three triangles are similar: $ABC \sim DEF$ with ratio $\frac{4}{3}$ and $ABC \sim GHI$ with ratio $\frac{4}{5}$ Check all ratios. Triangle $ABC$ has sides $12, 16, 20$ (ratio $3:4:5$). Triangle $DEF$ has sides $9, 12, 15$ (ratio $3:4:5$). Triangle $GHI$ has sides $15, 20, 25$ (ratio $3:4:5$). All three triangles have sides in the ratio $3:4:5$, so all are similar by SSS. The similarity ratios are: $\frac{AB}{DE} = \frac{12}{9} = \frac{4}{3}$ and $\frac{AB}{GH} = \frac{12}{15} = \frac{4}{5}$. Choice A is correct about similarity but vague about ratios. Choices B and C incorrectly exclude one triangle each.
15. Triangle $RST$ has $RS = 30$, $ST = 40$, and $\angle S = 90°$. Triangle $UVW$ has $UV = 18$, $VW = 24$, and $\angle V = 90°$. A student claims these are similar by SAS. What is the most complete analysis of this claim?
    Answer: The claim is correct because $\frac{RS}{UV} = \frac{ST}{VW} = \frac{5}{3}$ and both have right angles at corresponding vertices When analyzing triangle similarity claims, you need to carefully check both the side ratios and angle correspondence. For SAS (Side-Angle-Side) similarity, you need two pairs of proportional sides with the included angle being congruent. Let's examine the given information systematically. In triangle RST: RS = 30, ST = 40, with a right angle at S. In triangle UVW: UV = 18, VW = 24, with a right angle at V. First, check the side ratios: $\frac{RS}{UV} = \frac{30}{18} = \frac{5}{3}$ and $\frac{ST}{VW} = \frac{40}{24} = \frac{5}{3}$. The ratios are equal, so these sides are proportional. Second, both triangles have right angles at vertices S and V respectively, which are the vertices between the given sides. Since the angle between the proportional sides is the same (90°), SAS similarity is confirmed. Answer A incorrectly suggests you need to calculate hypotenuses to verify SAS similarity. While the calculations are correct (RT = 50, UW = 30), SAS only requires two sides and the included angle. Answer B makes the same error, incorrectly claiming you need the third side. Answer D misunderstands angle correspondence—the right angles are indeed between the corresponding sides we're comparing, making them the included angles needed for SAS. Remember: For SAS similarity, focus on whether you have two proportional sides with the included angle being congruent. Don't overcomplicate by checking unnecessary information like third sides or non-included angles.
16. In triangle $XYZ$, $XY = 21$, $YZ = 28$, and $\angle Y = 58°$. In triangle $RST$, $RS = 15$, $ST = 20$, and $\angle S = 58°$. Which statement about SAS similarity is correct?
    Answer: SAS similarity does not apply because we cannot determine if the angles are between corresponding proportional sides For SAS similarity, we need two pairs of proportional sides with the included angle equal. The ratios are $\frac{XY}{RS} = \frac{21}{15} = \frac{7}{5}$ and $\frac{YZ}{ST} = \frac{28}{20} = \frac{7}{5}$, which are equal. However, we don't know which sides in each triangle correspond to each other. Without knowing the correspondence, we cannot confirm that $\angle Y$ and $\angle S$ are the included angles between the proportional sides. Choices A and B assume correspondence and have calculation errors. Choice D has correct calculations but ignores the correspondence issue.

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