Similarity & Congruence - ACT Math

1. In $\triangle ABC$ and $\triangle DEF$, $AB=9$, $AC=12$, and the included angle $\angle A$ equals the included angle $\angle D$. Also, $DE=12$ and $DF=16$. Are the triangles similar? If so, why?
   Answer: Yes, by SAS similarity because $\frac{AB}{DE}=\frac{AC}{DF}$ and the included angles are equal. The triangles are similar by SAS similarity criterion. We have two sides and the included angle: AB/DE = 9/12 = 3/4 and AC/DF = 12/16 = 3/4. Since both ratios are equal (3/4) and the included angles (angle A and angle D) are equal, the triangles satisfy SAS similarity. The common ratio 3/4 is the scale factor from triangle ABC to triangle DEF.
2. Two triangles $\triangle PQR$ and $\triangle STU$ are shown. The sides with one tick mark are $PQ=4$ and $ST=6$, and the sides with two tick marks are $PR=10$ and $SU=15$. The included angles $\angle P$ and $\angle S$ are marked congruent with matching arcs. Are the triangles similar? If so, why?
   Answer: Yes; by SAS similarity. The triangles are similar by SAS similarity since two pairs of corresponding sides are proportional and the included angles are congruent. The sides $PQ = 4$ and $ST = 6$ give ratio $4/6 = 2/3$, while $PR = 10$ and $SU = 15$ give ratio $10/15 = 2/3$. Since the ratios are equal and the included angles $∠P ≅ ∠S$, the triangles are similar by SAS. This confirms that all corresponding sides will be proportional with the same scale factor.
3. Triangles $\triangle GHI$ and $\triangle JKL$ are shown. In $\triangle GHI$, $GH=9$, $HI=12$, and $GI=15$. In $\triangle JKL$, $JK=6$, $KL=8$, and $JL=10$. What is the scale factor from $\triangle JKL$ to $\triangle GHI$?
   Answer: $\dfrac{3}{2}$ The triangles are similar by SSS similarity because all corresponding sides are proportional. Assuming correspondence based on side lengths, JK to GH, KL to HI, and JL to GI. The ratios are GH/JK = 9/6 = 3/2, HI/KL = 12/8 = 3/2, and GI/JL = 15/10 = 3/2, confirming the scale factor from \triangle JKL to \triangle GHI is 3/2. This means sides in \triangle GHI are 3/2 times those in \triangle JKL. A common error is reversing the scale factor to 2/3, but the direction is from JKL to GHI.
4. Triangles $\triangle ABC$ and $\triangle DEF$ are shown with side lengths: $AB=9$, $BC=12$, $AC=15$ and $DE=6$, $EF=8$, $DF=10$. What is the scale factor from $\triangle DEF$ to $\triangle ABC$?
   Answer: $\dfrac{3}{2}$ Triangles ABC and DEF are similar by the SSS similarity criterion because all three pairs of corresponding sides are proportional. Assuming correspondences A to D, B to E, and C to F, the side ratios are $AB/DE = 9/6 = 3/2$, $BC/EF = 12/8 = 3/2$, and $AC/DF = 15/10 = 3/2$. Thus, the scale factor from DEF to ABC is $3/2$. This consistent ratio confirms enlargement from the smaller to the larger triangle.
5. Two triangles, $\triangle DEF$ and $\triangle GHI$, are congruent. If $DE = 5$, $EF = 12$, $DF = 13$, what is the length of $GH$ if $GH$ corresponds to $DE$?
   Answer: 5 Since the triangles are congruent, all corresponding sides are equal in length. Triangle DEF has sides DE = 5, EF = 12, and DF = 13. In congruent triangles, corresponding sides have identical measurements. If GH corresponds to DE, then GH = DE = 5.
6. Two triangles $\triangle ABC$ and $\triangle DEF$ are shown. $AB=5$, $AC=7$, and $BC=8$. In the other triangle, $DE=10$, $DF=14$, and $EF=16$. Which statement is correct?
   Answer: The triangles are similar with scale factor $2$ from $\triangle ABC$ to $\triangle DEF$. The triangles are similar by SSS since all corresponding sides are proportional with the same scale factor. For triangle ABC with sides 5, 7, 8 and triangle DEF with sides 10, 14, 16, the ratios are: 10/5 \= 2, 14/7 \= 2, and 16/8 \= 2. Since all three ratios equal 2, the triangles are similar with scale factor 2 from triangle ABC to triangle DEF. This means each side of triangle DEF is twice the length of the corresponding side in triangle ABC.
7. Triangles WXY and ZAB are shown to be similar by which criterion if $\text{angle } W = \text{angle } Z$ and $\text{angle } X = \text{angle } A$?
   Answer: AA Triangles WXY and ZAB are similar by AA (Angle-Angle) criterion. We have two pairs of equal angles: angle W = angle Z and angle X = angle A. When two angles of one triangle are equal to two angles of another triangle, the third angles must also be equal by the angle sum property. This is sufficient to prove similarity using the AA criterion.
8. Triangles $\triangle ABC$ and $\triangle DEF$ are shown. The side markings indicate $AB \cong DE$ (one tick) and $BC \cong EF$ (two ticks). Also, $\angle B \cong \angle E$ is marked, and it is the included angle between the tick-marked sides. Which congruence criterion applies?
   Answer: SAS The triangles are congruent by SAS since two pairs of corresponding sides are congruent (AB ≅ DE and BC ≅ EF) and the included angle between them is congruent (∠B ≅ ∠E). The SAS criterion requires that the angle be between the two marked sides. Since angle B is between sides AB and BC, and angle E is between sides DE and EF, the SAS criterion is satisfied.
9. In $\triangle ABC$, $AB=7$, $AC=9$, and $BC=12$. In $\triangle DEF$, $DE=14$, $DF=18$, and $EF=20$. Are the triangles similar? If so, why?
   Answer: No; the three side ratios are not all equal To check similarity by SSS, all three side ratios must be equal. For △ABC (7, 9, 12) and △DEF (14, 18, 20), we calculate: DE/AB = 14/7 = 2, DF/AC = 18/9 = 2, but EF/BC = 20/12 = 5/3 ≈ 1.67. Since 2 ≠ 5/3, the ratios are not all equal. Therefore, the triangles are not similar because the three side ratios are not equal.
10. Two triangles, $\triangle JKL$ and $\triangle MNO$, are congruent. If $JK = 8$, $KL = 15$, $JL = 17$, what is the length of $MN$ if $MN$ corresponds to $JK$?
    Answer: 8 Since the triangles are congruent, all corresponding sides are equal in length. Triangle JKL has sides JK = 8, KL = 15, and JL = 17. In congruent triangles, corresponding sides have identical measurements. If MN corresponds to JK, then MN = JK = 8.
11. Two triangles have side lengths $\triangle ABC$: $AB=6$, $BC=10$, $AC=12$ and $\triangle DEF$: $DE=3$, $EF=5$, $DF=6$. Are the triangles similar? If so, why?
    Answer: Yes; SSS similarity because all corresponding side ratios are equal. The triangles are similar by SSS similarity because all corresponding side ratios are equal. Assuming correspondence based on side lengths, AB to DE, BC to EF, and AC to DF, with ratios DE/AB = 3/6 = 1/2, EF/BC = 5/10 = 1/2, and DF/AC = 6/12 = 1/2. These consistent proportions confirm similarity. They are not congruent because the sides are not equal, only proportional. A key distractor is mistaking proportionality for congruence, but the sides differ in length.
12. Two triangles have side lengths $8$, $10$, $12$ and $12$, $15$, $18$. Which triangles are similar (using the given side lengths only)?
    Answer: Similar by SSS, scale factor $\frac{3}{2}$ from the $8$-$10$-$12$ triangle to the $12$-$15$-$18$ triangle. The triangles are similar by SSS similarity criterion. Checking the ratios: 12/8 = 3/2, 15/10 = 3/2, and 18/12 = 3/2. Since all three ratios equal 3/2, the corresponding sides are proportional. The scale factor from the 8-10-12 triangle to the 12-15-18 triangle is 3/2, confirming they are similar triangles.
13. In similar triangles LMN and OPQ, if the scale factor from LMN to OPQ is $3:2$ and $LM = 9$, what is the length of $OP$?
    Answer: 6 In similar triangles LMN and OPQ, the scale factor from LMN to OPQ is 3:2, meaning sides of OPQ are 2/3 the length of corresponding sides in LMN. The sides LM and OP correspond to each other, with LM = 9. Setting up the proportion: OP/LM = 2/3, we get OP/9 = 2/3. Cross-multiplying: 3 × OP = 2 × 9, so 3 × OP = 18, giving OP = 6.
14. In $\triangle XYZ$ and $\triangle ABC$, $\triangle XYZ$ has angles $X = 90^\text{o}$, $Y = 45^\text{o}$, and $\triangle ABC$ has angles $A = 90^\text{o}$, $B = 45^\text{o}$. Are the triangles similar?
    Answer: Yes, by AA similarity. The triangles are similar by AA similarity criterion because they have two pairs of equal angles. Triangle XYZ has angles 90°, 45°, and 45° (since angles sum to 180°). Triangle ABC has angles 90°, 45°, and 45° (since angles sum to 180°). Having two pairs of equal angles (90° = 90° and 45° = 45°) confirms similarity by AA criterion.
15. Two triangles are shown with side lengths. Triangle $\triangle GHI$ has $GH=7$, $HI=9$, and $GI=12$. Triangle $\triangle JKL$ has $JK=14$, $KL=18$, and $JL=20$. Are the triangles similar? If so, why?
    Answer: No; the side ratios are not all equal. Triangles GHI and JKL are not similar because the corresponding sides are not proportional. Assuming correspondences G to J, H to K, and I to L based on side lengths, the ratios are JK/GH = 14/7 = 2, KL/HI = 18/9 = 2, but JL/GI = 20/12 = 5/3, which are inconsistent. Without a constant scale factor, similarity criteria like SSS or SAS are not met. This addresses the distractor that unequal ratios still allow similarity, confirming they do not.
16. Triangles $\triangle ABC$ and $\triangle DEF$ are similar with correspondence $A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$. If $AB=9$, $BC=12$, $AC=15$, and $DE=6$, what is the length of $EF$?
    Answer: $8$ The triangles are similar with correspondence A↔D, B↔E, C↔F, meaning AB corresponds to DE, BC corresponds to EF, and AC corresponds to DF. First, find the scale factor using the known corresponding sides: DE/AB = 6/9 = 2/3. Since the triangles are similar, all corresponding sides have the same ratio. To find EF, we use the proportion: EF/BC = 2/3. Therefore, EF = BC × (2/3) = 12 × (2/3) = 8. The length of EF is 8.
17. Which congruence criterion applies to triangles ABC and DEF if $AB = DE$, $\text{angle } B = \text{angle } E$, and $BC = EF$?
    Answer: SAS The triangles are congruent by SAS (Side-Angle-Side) criterion. We have two pairs of equal sides: AB = DE and BC = EF, with the included angle B equal to angle E. The SAS criterion states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
18. Two triangles are shown. In $\triangle ABC$, $AB=6$, $BC=9$, $AC=12$. In $\triangle DEF$, $DE=4$, $EF=6$, $DF=8$. Are the triangles similar? If so, why?
    Answer: Yes; by SSS similarity. The triangles are similar by SSS since all three pairs of corresponding sides are proportional. For triangle ABC with sides 6, 9, 12 and triangle DEF with sides 4, 6, 8, the ratios are: 4/6 \= 2/3, 6/9 \= 2/3, and 8/12 \= 2/3. Since all three ratios equal 2/3, the corresponding sides are proportional with scale factor 2/3. This satisfies the SSS similarity criterion.
19. Two triangles, $\triangle ABC$ and $\triangle DEF$, are shown with $\triangle ABC \text{ having sides } AB = 6, BC = 8, \text{ and } AC = 10$. $\triangle DEF \text{ has sides } DE = 9, EF = 12, \text{ and } DF = 15$. Are the triangles similar? If so, why?
    Answer: Yes, by SSS similarity. The triangles are similar by SSS similarity criterion because all corresponding sides are proportional. Triangle ABC has sides 6, 8, 10 and triangle DEF has sides 9, 12, 15. The ratios are 9/6 = 3/2, 12/8 = 3/2, and 15/10 = 3/2. Since all three ratios are equal (3/2), the triangles are similar with a scale factor of 3/2.
20. Triangles $\triangle RST$ and $\triangle XYZ$ are similar. The scale factor from $\triangle RST$ to $\triangle XYZ$ is $\dfrac{3}{2}$. If $RS=10$ and $ST=14$, what is the perimeter of $\triangle XYZ$ if the perimeter of $\triangle RST$ is $36$?
    Answer: $54$ The triangles are similar with a scale factor of 3/2 from triangle RST to triangle XYZ. This means each side of triangle XYZ is 3/2 times the corresponding side of triangle RST. Since similarity preserves ratios, the perimeter of triangle XYZ is also 3/2 times the perimeter of triangle RST. Given that the perimeter of triangle RST is 36, the perimeter of triangle XYZ = 36 × (3/2) = 54. We can verify: if RS = 10 and ST = 14, and the perimeter is 36, then RT = 12, and the scaled sides would be 15, 21, and 18, which sum to 54.

Start this quiz