ACT Math Quiz: Similarity And Congruence

20 questions · exam conditions

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Similarity And CongruenceQuestion 1 of 20

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

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

ASA

AAS

SAS

SSS

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ACT Math Quiz

ACT Math Quiz: Similarity And Congruence

Practice Similarity And Congruence in ACT Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

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

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.

All questions

Question 1

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

ASA (correct answer)
AAS
SAS
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 $\triangle ABC$ , $AB=6$ , $AC=9$ , $BC=12$ . In $\triangle DEF$ , $DE=4$ , $DF=6$ , $EF=8$ . What is the scale factor from $\triangle ABC$ to $\triangle DEF$ (i.e., multiply lengths in $\triangle ABC$ by what number to get corresponding lengths in $\triangle DEF$ )?

$\dfrac{3}{2}$
$\dfrac{2}{3}$ (correct answer)
$\dfrac{4}{3}$
$\dfrac{1}{2}$

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 $\triangle JKL$ and $\triangle MNO$ are similar. Corresponding sides are $JK \leftrightarrow MN$ , $KL \leftrightarrow NO$ , and $JL \leftrightarrow MO$ . If $JK=8$ , $KL=10$ , $JL=12$ , and $MN=12$ , what is the length of $NO$ ?

$12$
$15$ (correct answer)
$\dfrac{20}{3}$
$\dfrac{25}{3}$

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

$10$ (correct answer)
$18$
$20$
$22.5$

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

$\dfrac{2}{3}$
$\dfrac{3}{2}$ (correct answer)
$\dfrac{5}{3}$
$\dfrac{3}{5}$

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 $\triangle XYZ$ and $\triangle ABC$ , $\triangle XYZ$ has angles $X = 45^\text{o}$ , $Y = 45^\text{o}$ , and $\triangle ABC$ has angles $A = 45^\text{o}$ , $B = 45^\text{o}$ . Are the triangles similar?

Yes, by SSS similarity.
No, they are not similar.
Yes, by SAS similarity.
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, $\triangle GHI$ and $\triangle JKL$ , $\triangle GHI$ has angles $G = 60^\text{o}$ , $H = 60^\text{o}$ , and $\triangle JKL$ has angles $J = 60^\text{o}$ , $K = 60^\text{o}$ . Are the triangles similar?

Yes, by SSS similarity.
No, they are not similar.
Yes, by AA similarity. (correct answer)
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 $\triangle ABC$ 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?

AAS
ASA
SSS
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 $\triangle ABC$ , $AB\=6$ , $BC\=9$ , $AC\=12$ . In $\triangle DEF$ , $DE\=4$ , $EF\=6$ , $DF\=8$ . Are the triangles similar? If so, why?

Yes; by ASA.
No; the scale factor is different for each pair of sides.
Yes; by SSS similarity. (correct answer)
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 $\triangle XYZ \thicksim \triangle VWU$ with $XZ = 6$ and $VW = 12$ , what is the ratio of $YZ$ to $WU$ ?

3:2
2:1
1:1
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 $XZ/VW = 6/12 = 1/2$ . Since the triangles are similar, all corresponding sides have the same ratio. Therefore, YZ corresponds to WU, and the ratio $YZ:WU = 1:2$ .

Question 11

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 $\triangle ABC$ ?

$\dfrac{2}{3}$
$\dfrac{3}{2}$ (correct answer)
$\dfrac{4}{5}$
$\dfrac{5}{4}$

Explanation: 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.

Question 12

Two triangles are shown. In $\triangle RST$ , angles at $R$ and $S$ are marked with one arc and two arcs, respectively. In $\triangle VWX$ , angles at $V$ and $W$ are marked with one arc and two arcs, respectively.

Which congruence/similarity criterion applies to conclude the triangles are similar?

SSS
SAS
AA (correct answer)
ASA

Explanation: Triangles RST and VWX are similar by the AA similarity criterion because they have two pairs of corresponding angles congruent, with angle R equal to angle V and angle S equal to angle W. The correspondences are R to V, S to W, and thus T to X. No side lengths are given, so similarity relies solely on the equal angles, and the third angles are also equal by the triangle angle sum. This criterion is sufficient for similarity but not for congruence, as sides may not be equal.

Question 13

Two triangles are drawn with the following markings: In $\triangle ABC$ and $\triangle DEF$ , sides $AB$ and $DE$ each have one tick mark, sides $AC$ and $DF$ each have two tick marks, and the included angles $\angle A$ and $\angle D$ are marked equal (single arc). The labeled lengths are $AB=5$ , $AC=7$ , $DE=5$ , and $DF=7$ .

Which congruence criterion applies to conclude $\triangle ABC \cong \triangle DEF$ ?

ASA
SSS
AAS
SAS (correct answer)

Explanation: The triangles are congruent by SAS congruence because two sides and the included angle are equal. The correspondences are side AB to DE (both 5 with one tick), side AC to DF (both 7 with two ticks), and included angle A to angle D (equal markings). With AB = DE, AC = DF, and angle A = angle D, the SAS criterion applies directly. All corresponding sides and angles are equal due to congruence. This distinguishes from AAS or ASA, as the equal angle is specifically included between the equal sides.

Question 14

Two triangles are shown with angle markings. In $\triangle PQR$ , $\angle P$ has one arc and $\angle Q$ has two arcs. In $\triangle STU$ , $\angle S$ has one arc and $\angle T$ has two arcs. Side lengths are not needed.

Are the triangles similar? If so, why?

Yes; $AA$ similarity because two corresponding angles are congruent. (correct answer)
Yes; $SSS$ similarity because all three sides are equal.
No; angle congruence alone cannot prove similarity.
Yes; $SAS$ similarity because two sides and the included angle are congruent.

Explanation: Triangles PQR and STU are similar by AA similarity because two pairs of corresponding angles are congruent, as indicated by the arc markings. The correspondences are angle P to angle S (both with one arc) and angle Q to angle T (both with two arcs), implying angle R corresponds to angle U by the triangle angle sum. Since two corresponding angles are equal, the triangles are similar without needing side lengths. This rules out SSS or SAS options, which incorrectly assume side equality or proportionality here, emphasizing that AA is sufficient for similarity.

Question 15

Two triangles are shown with side tick marks and one angle mark. In $\triangle JKL$ and $\triangle MNO$ , $JK$ and $MN$ each have one tick mark, $JL$ and $MO$ each have two tick marks, and the included angles $\angle J$ and $\angle M$ are each marked with a single arc.

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

ASA
SSS
SAS (correct answer)
AAS

Explanation: Triangles JKL and MNO are congruent by SAS congruence because two pairs of corresponding sides are equal and the included angles are equal. The correspondences are side JK to MN (one tick each), side JL to MO (two ticks each), and included angle J to angle M (single arc each). With the equal sides surrounding the equal included angle, all corresponding sides and angles are equal. This distinguishes SAS from SSS, ASA, or AAS, which do not match the given equal parts.

Question 16

Two triangles have $\angle A \cong \angle D$ , $\angle C \cong \angle F$ , and the side between those angles satisfies $AC=7$ and $DF=7$ . Which congruence criterion applies?

SSS
SAS
ASA (correct answer)
AAS

Explanation: The triangles are congruent by ASA (Angle-Side-Angle) criterion. We have angle A congruent to angle D, angle C congruent to angle F, and the side between these angles (AC and DF) are equal at 7 units. ASA requires two angles and the included side between them to be congruent, which is exactly what we have here.

Question 17

Triangle DEF is similar to triangle GHI. If $DE = 6$ , $EF = 8$ , and $GH = 12$ , what is the length of $HI$ ?

16 (correct answer)
10
18
20

Explanation: Triangles DEF and GHI are similar, meaning their corresponding sides are proportional. The sides DE and GH correspond to each other, with DE = 6 and GH = 12, giving a scale factor of 12/6 = 2/1. The sides EF and HI also correspond, with EF = 8. Setting up the proportion: HI/EF = GH/DE, we get HI/8 = 12/6 = 2, so HI = 8 × 2 = 16.

Question 18

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?

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

Explanation: 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.

Question 19

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?

No; two sides are proportional but the included angles are not equal.
Yes; by SSS similarity.
Yes; by SAS similarity. (correct answer)
No; SAS requires all three sides to be proportional.

Explanation: 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.

Question 20

Triangles $\triangle ABC$ and $\triangle DEF$ are shown. $\angle A \cong \angle D$ and $\angle B \cong \angle E$ are marked. Side $AB\=10$ and $DE\=5$ . What is the scale factor from $\triangle ABC$ to $\triangle DEF$ ?

$2$
$\dfrac{1}{2}$ (correct answer)
$5$
$\dfrac{5}{2}$

Explanation: The triangles are similar by AA since two pairs of corresponding angles are congruent: ∠A ≅ ∠D and ∠B ≅ ∠E. The scale factor from triangle ABC to triangle DEF is the ratio of corresponding sides DE to AB, which is 5/10 = 1/2. This means each side of triangle DEF is half the length of the corresponding side in triangle ABC.