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ACT Math Help: Similarity And Congruence

Review real example questions for Similarity And Congruence in ACT Math.

Question 1 / 10

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Two triangles are shown with markings indicating equal parts. In △ABC\triangle ABC and △DEF\triangle DEF, ∠A\angle A is marked congruent to ∠D\angle D (one arc), and ∠B\angle B is marked congruent to ∠E\angle E (two arcs). The side between those angles, ABAB, has one tick mark, and the corresponding side DEDE also has one tick mark.

Which congruence criterion applies (SSS, SAS, ASA, AAS)?

All questions

Question 1

Two triangles are shown with markings indicating equal parts. In △ABC\triangle ABC and △DEF\triangle DEF, ∠A\angle A is marked congruent to ∠D\angle D (one arc), and ∠B\angle B is marked congruent to ∠E\angle E (two arcs). The side between those angles, ABAB, has one tick mark, and the corresponding side DEDE also has one tick mark.

Which congruence criterion applies (SSS, SAS, ASA, AAS)?

  1. ASA (correct answer)
  2. AAS
  3. SAS
  4. SSS

Explanation: Triangles ABC and DEF are congruent by ASA congruence because two pairs of corresponding angles are equal and the included sides are equal. The correspondences are angle A to angle D (one arc) and angle B to angle E (two arcs), with included side AB to DE (one tick each). With the equal angles surrounding the equal included side, all corresponding parts are equal. This distinguishes ASA from AAS, which involves a non-included side, emphasizing the importance of the side's position.

Question 2

Two triangles are shown. In △ABC\triangle ABC, AB=6AB=6, AC=9AC=9, BC=12BC=12. In △DEF\triangle DEF, DE=4DE=4, DF=6DF=6, EF=8EF=8. What is the scale factor from △ABC\triangle ABC to △DEF\triangle DEF (i.e., multiply lengths in △ABC\triangle ABC by what number to get corresponding lengths in △DEF\triangle DEF)?

  1. 32\dfrac{3}{2}
  2. 23\dfrac{2}{3} (correct answer)
  3. 43\dfrac{4}{3}
  4. 12\dfrac{1}{2}

Explanation: To find the scale factor from triangle ABC to triangle DEF, we need to check if the triangles are similar by comparing ratios of corresponding sides. Let's check: DE/AB = 4/6 = 2/3, DF/AC = 6/9 = 2/3, and EF/BC = 8/12 = 2/3. Since all three ratios are equal, the triangles are similar by SSS similarity. The scale factor from triangle ABC to triangle DEF is 2/3, meaning we multiply each side length in triangle ABC by 2/3 to get the corresponding side length in triangle DEF.

Question 3

Triangles △JKL\triangle JKL and △MNO\triangle MNO are similar. Corresponding sides are JK↔MNJK \leftrightarrow MN, KL↔NOKL \leftrightarrow NO, and JL↔MOJL \leftrightarrow MO. If JK=8JK=8, KL=10KL=10, JL=12JL=12, and MN=12MN=12, what is the length of NONO?

  1. 1212
  2. 1515 (correct answer)
  3. 203\dfrac{20}{3}
  4. 253\dfrac{25}{3}

Explanation: The triangles are similar with given correspondences: JK ↔ MN, KL ↔ NO, and JL ↔ MO. First, find the scale factor using the known corresponding sides: MN/JK = 12/8 = 3/2. Since the triangles are similar, all corresponding sides have the same ratio. To find NO, we use the proportion: NO/KL = 3/2. Therefore, NO = KL × (3/2) = 10 × (3/2) = 15. The length of NO is 15.

Question 4

Triangles △GHI\triangle GHI and △JKL\triangle JKL are similar by AA with correspondence G↔JG\leftrightarrow J, H↔KH\leftrightarrow K, I↔LI\leftrightarrow L. If GH=12GH=12, JK=8JK=8, and HI=15HI=15, what is the length of the corresponding side KLKL?

  1. 1010 (correct answer)
  2. 1818
  3. 2020
  4. 22.522.5

Explanation: The triangles are similar by AA with G↔J, H↔K, I↔L, so HI corresponds to KL. The scale factor from △GHI to △JKL is JK/GH = 8/12 = 2/3. Therefore, KL = HI × scale factor = 15 × (2/3) = 10. Note that we're scaling down from the larger to the smaller triangle, so we multiply by 2/3.

Question 5

Triangles △PQR\triangle PQR and △STU\triangle STU are similar by AA. Angle ∠P\angle P corresponds to ∠S\angle S, and ∠Q\angle Q corresponds to ∠T\angle T. If PQ=6PQ=6 and the corresponding side ST=9ST=9, what is the scale factor from △PQR\triangle PQR to △STU\triangle STU?

  1. 23\dfrac{2}{3}
  2. 32\dfrac{3}{2} (correct answer)
  3. 53\dfrac{5}{3}
  4. 35\dfrac{3}{5}

Explanation: The triangles are similar by AA, with P↔S and Q↔T, so side PQ corresponds to side ST. The scale factor from △PQR to △STU is the ratio of corresponding sides: ST/PQ = 9/6 = 3/2. This means each side of △STU is 3/2 times the corresponding side of △PQR. The scale factor is 3/2, not 2/3, because we're scaling from the smaller to the larger triangle.

Question 6

For triangles △XYZ\triangle XYZ and △ABC\triangle ABC, △XYZ\triangle XYZ has angles X=45oX = 45^\text{o}, Y=45oY = 45^\text{o}, and △ABC\triangle ABC has angles A=45oA = 45^\text{o}, B=45oB = 45^\text{o}. Are the triangles similar?

  1. Yes, by SSS similarity.
  2. No, they are not similar.
  3. Yes, by SAS similarity.
  4. Yes, by AA similarity. (correct answer)

Explanation: The triangles are similar by AA similarity criterion because they have two pairs of equal angles. Triangle XYZ has angles 45°, 45°, and 90° (since angles sum to 180°). Triangle ABC has angles 45°, 45°, and 90° (since angles sum to 180°). Having two pairs of equal angles (45° = 45° and 45° = 45°) confirms similarity by AA criterion.

Question 7

Two triangles, △GHI\triangle GHI and △JKL\triangle JKL, △GHI\triangle GHI has angles G=60oG = 60^\text{o}, H=60oH = 60^\text{o}, and △JKL\triangle JKL has angles J=60oJ = 60^\text{o}, K=60oK = 60^\text{o}. Are the triangles similar?

  1. Yes, by SSS similarity.
  2. No, they are not similar.
  3. Yes, by AA similarity. (correct answer)
  4. Yes, by SAS similarity.

Explanation: The triangles are similar by AA similarity criterion because they have two pairs of equal angles. Triangle GHI has angles 60°, 60°, and 60° (since it's equilateral with angles summing to 180°). Triangle JKL has angles 60°, 60°, and 60° (since it's equilateral with angles summing to 180°). Having two pairs of equal angles (60° = 60° and 60° = 60°) confirms similarity by AA criterion.

Question 8

Triangles △ABC\triangle ABC and △DEF\triangle DEF are shown. The side markings indicate AB≅DEAB \cong DE (one tick) and BC≅EFBC \cong EF (two ticks). Also, ∠B≅∠E\angle B \cong \angle E is marked, and it is the included angle between the tick-marked sides. Which congruence criterion applies?

  1. AAS
  2. ASA
  3. SSS
  4. SAS (correct answer)

Explanation: 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.

Question 9

Two triangles are shown. In △ABC\triangle ABC, AB6ˉAB\=6, BC9ˉBC\=9, AC1ˉ2AC\=12. In △DEF\triangle DEF, DE4ˉDE\=4, EF6ˉEF\=6, DF8ˉDF\=8. Are the triangles similar? If so, why?

  1. Yes; by ASA.
  2. No; the scale factor is different for each pair of sides.
  3. Yes; by SSS similarity. (correct answer)
  4. No; triangles with different perimeters cannot be similar.

Explanation: 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.

Question 10

If △XYZ∼△VWU\triangle XYZ \thicksim \triangle VWU with XZ=6XZ = 6 and VW=12VW = 12, what is the ratio of YZYZ to WUWU?

  1. 3:2
  2. 2:1
  3. 1:1
  4. 1:2 (correct answer)

Explanation: Triangles XYZ and VWU are similar, meaning corresponding sides are proportional. However, the correspondence order matters: XZ corresponds to VW (not VU), so XZ/VW=6/12=1/2XZ/VW = 6/12 = 1/2. Since the triangles are similar, all corresponding sides have the same ratio. Therefore, YZ corresponds to WU, and the ratio YZ:WU=1:2YZ:WU = 1:2.