Similarity & Congruence
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ACT Math › Similarity & Congruence
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)?
AAS
SAS
SSS
ASA
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.
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 $\triangle DEF$)?
$\dfrac{2}{3}$
$\dfrac{3}{2}$
$\dfrac{1}{2}$
$\dfrac{4}{3}$
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.
In $\triangle ABC$ and $\triangle DEF$, $\angle A\cong\angle D$, $\angle B\cong\angle E$, and the side between them is equal: $AB=DE=7$. Which congruence criterion applies (SSS, SAS, ASA, AAS)?
SSS
SAS
ASA
AAS
Explanation
The triangles are congruent by ASA (Angle-Side-Angle) criterion. We have angle A ≅ angle D, angle B ≅ angle E, and the side between these angles is equal: AB = DE = 7. In ASA congruence, we need two angles and the included side (the side between the two angles) to be equal. Since AB is the side between angles A and B, and DE is the side between angles D and E, and these sides are equal along with their adjacent angles, the triangles are congruent by ASA.
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 $NO$?
$12$
$15$
$\dfrac{20}{3}$
$\dfrac{25}{3}$
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.
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 $KL$?
$22.5$
$10$
$18$
$20$
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.
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 $\triangle STU$?
$\dfrac{3}{2}$
$\dfrac{3}{5}$
$\dfrac{5}{3}$
$\dfrac{2}{3}$
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.
In triangle ABC and triangle XYZ, if $\text{angle } A = \text{angle } X$ and $\text{angle } B = \text{angle } Y$, are the triangles similar?
Yes, by SAS
Yes, by AA
No
Yes, by SSS
Explanation
The triangles are similar by AA (Angle-Angle) criterion. We have two pairs of equal angles: angle A = angle X and angle B = angle Y. 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.
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?
No, they are not similar.
Yes, by SAS similarity.
Yes, by SSS similarity.
Yes, by AA similarity.
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.
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 AA similarity.
Yes, by SSS similarity.
Yes, by SAS similarity.
No, they are not similar.
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.
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?
SSS
SAS
ASA
AAS
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.