Geometry Quiz: Aa Criterion From Similarity Transformations

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Aa Criterion From Similarity TransformationsQuestion 1 of 20

Two triangles, MNP and QRS, are being tested for similarity. It is known that $\angle M = \angle Q = 48°$ and that triangle QRS can be obtained from triangle MNP through a sequence of similarity transformations. A student concludes that this information alone is sufficient to prove similarity using the AA criterion. What is the error in this reasoning?

The AA criterion requires two pairs of equal corresponding angles, but only one pair is confirmed from the given information

Similarity transformations do not guarantee that corresponding angles remain equal, so additional verification is needed for the conclusion

The student correctly applied the AA criterion, but failed to verify that the similarity transformations preserve triangle orientation

The conclusion is valid since similarity transformations automatically ensure that all corresponding angle pairs are equal by definition

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Geometry Quiz

Geometry Quiz: Aa Criterion From Similarity Transformations

Practice Aa Criterion From Similarity Transformations in Geometry 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 Aa Criterion From Similarity Transformations, giving you a quick way to practice the rules, question types, and explanations that matter most for Geometry.

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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.

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Question 1

The AA criterion requires two pairs of equal corresponding angles, but only one pair is confirmed from the given information (correct answer)
Similarity transformations do not guarantee that corresponding angles remain equal, so additional verification is needed for the conclusion
The student correctly applied the AA criterion, but failed to verify that the similarity transformations preserve triangle orientation
The conclusion is valid since similarity transformations automatically ensure that all corresponding angle pairs are equal by definition

Explanation: The AA criterion requires two pairs of equal corresponding angles to establish similarity. While we know

\angle M = \angle Q = 48°

, we need information about a second pair of corresponding angles to apply AA criterion. The fact that one triangle can be obtained from another through similarity transformations would indeed guarantee similarity, but the student's reasoning specifically claims to use the AA criterion, which requires two angle pairs. Choice B incorrectly states that similarity transformations don't preserve angles. Choice C focuses on irrelevant orientation issues. Choice D would be correct if the student claimed similarity from the transformations, but not for AA criterion application.

Question 2

Triangle JKL undergoes a similarity transformation consisting of a reflection across line m followed by a dilation with scale factor k > 0. The resulting triangle J'K'L' has $\angle J' = 42°$ and $\angle K' = 85°$ . If a student wants to prove that triangles JKL and J'K'L' are similar using the AA criterion, what information about triangle JKL is sufficient?

Knowing that $\angle J = 42°$ and $\angle K = 85°$ , since similarity transformations preserve all angle measures completely (correct answer)
Knowing that $\angle J = 42°$ and $\angle K = 85°$ , plus verification that the reflection line and dilation center are properly positioned
Knowing any two angle measures in triangle JKL, since the transformation process guarantees that corresponding angles will be equal
Knowing that $\angle J = 42°$ and $\angle K = 85°$ , plus confirmation that the scale factor k produces proportional side lengths

Explanation: Similarity transformations always preserve angle measures, so if

\angle J' = 42°

and

\angle K' = 85°

in the image triangle, then

\angle J = 42°

and

\angle K = 85°

in the original triangle. This gives us two pairs of equal corresponding angles, satisfying the AA criterion. Choice B incorrectly suggests that geometric positioning affects angle preservation. Choice C is wrong because we need the specific angles to match, not just any two angles. Choice D adds unnecessary verification since proportional sides are guaranteed by the AA criterion establishing similarity.

Question 3

A dilation followed by a rotation is used to map triangle $\triangle UVW$ to triangle $\triangle U'V'W'$ . In the diagram, $\angle U \cong \angle U'$ is marked with one arc and $\angle V \cong \angle V'$ is marked with two arcs. No side lengths are labeled, and the diagram is not drawn to scale. Which conclusion about the triangles is valid?

$\triangle UVW \sim \triangle U'V'W'$ by AA, so $\dfrac{UV}{U'V'}=\dfrac{VW}{V'W'}$ . (correct answer)
$\triangle UVW \cong \triangle U'V'W'$ because a rotation preserves lengths.
$UV=U'V'$ because $\angle U \cong \angle U'$ and $\angle V \cong \angle V'$ .
$\triangle UVW \sim \triangle V'U'W'$ because $\angle U \cong \angle V'$ and $\angle V \cong \angle U'$ .

Explanation: The skill being assessed is the AA criterion for triangle similarity. The AA criterion states that if two pairs of corresponding angles in two triangles are congruent, then the triangles are similar, as a dilation followed by a rotation can map one to the other while maintaining angle measures. In this problem, the marked angles are angle U congruent to angle U' with one arc and angle V congruent to angle V' with two arcs. Applying the AA criterion, triangle UVW is similar to triangle U'V'W' with correspondence U to U', V to V', and W to W'. Because the triangles are similar, their corresponding sides are proportional, meaning the ratios of the lengths of corresponding sides are equal, but the sides themselves are not necessarily equal in length. A common misconception is to assume a rotation alone implies equal sides, but the dilation component allows for different scales. When approaching similar problems, always check for matching angles first to establish similarity before comparing side lengths or ratios.

Question 4

Triangle ABC has angles measuring 45°, 60°, and 75°. Triangle DEF undergoes a sequence of similarity transformations (rotation, reflection, and dilation with scale factor 3) to produce triangle GHI. If triangle GHI has angles measuring 45°, 60°, and 75°, which statement about the relationship between triangles ABC and GHI is most accurate?

The triangles are similar by AA criterion, and the transformations preserve angle measures while scaling side lengths proportionally (correct answer)
The triangles are congruent because similarity transformations always preserve both angle measures and side lengths exactly
The triangles are not similar because the dilation changes the side lengths, violating the definition of similarity
The triangles are similar only if the rotation and reflection map corresponding vertices to the same relative positions

Explanation: Since similarity transformations preserve angle measures, triangles ABC and GHI have all corresponding angles equal (45°, 60°, 75°). By the AA criterion, having two pairs of equal corresponding angles guarantees similarity (the third pair is automatically equal). The dilation scales all sides proportionally while preserving angles, which is exactly what defines similarity. Choice B is wrong because similarity transformations don't preserve side lengths exactly (only proportionally). Choice C incorrectly suggests that changing side lengths violates similarity. Choice D incorrectly focuses on vertex positioning rather than angle equality.

Question 5

In the plane, triangles $\triangle LMN$ and $\triangle QRS$ are drawn. The diagram marks $\angle L \cong \angle Q$ with one arc and $\angle M \cong \angle R$ with two arcs. No side lengths are shown, and the diagram is not drawn to scale. Which statement proves the triangles are similar?

The triangles are similar because they look like the same shape in the diagram.
The triangles are congruent because two corresponding angles are marked congruent.
The triangles are similar by AA because two pairs of corresponding angles are marked congruent. (correct answer)
The triangles are similar because $LM=QR$ and $MN=RS$ .

Explanation: The skill being assessed is the AA criterion for triangle similarity. The AA criterion states that if two pairs of corresponding angles in two triangles are congruent, then the triangles are similar, which can be verified through similarity transformations preserving angles. In this problem, the marked angles are angle L congruent to angle Q with one arc and angle M congruent to angle R with two arcs. Applying the AA criterion, triangle LMN is similar to triangle QRS with correspondence L to Q, M to R, and N to S. Because the triangles are similar, their corresponding sides are proportional, meaning the ratios of the lengths of corresponding sides are equal, but the sides themselves are not necessarily equal in length. A common misconception is to rely on visual appearance for similarity without confirming angle congruences, but diagrams not to scale require explicit markings. When approaching similar problems, always check for matching angles first to establish similarity before comparing side lengths or ratios.

Question 6

A triangle $\triangle ABC$ is drawn on the left. On the right, a triangle $\triangle A'B'C'$ is drawn as if it could be obtained by dilating $\triangle ABC$ and then applying a rigid motion. Angle markings show $\angle B \cong \angle B'$ (single arc) and $\angle C \cong \angle C'$ (double arc). No side lengths are labeled, and the diagram is not drawn to scale.

Which reasoning uses similarity transformations correctly?

A dilation followed by a rigid motion can map one triangle to the other, so they are similar by AA. (correct answer)
A rigid motion alone maps one triangle to the other, so they are congruent.
Since two angles match, corresponding sides must be equal.
Because the triangles are not drawn to scale, no conclusion can be made.

Explanation: The AA criterion is a key method for proving triangle similarity. If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. In this diagram, angle B is marked congruent to angle B' with a single arc, and angle C is marked congruent to angle C' with a double arc. Therefore, by the AA similarity criterion, triangle ABC is similar to triangle A'B'C'. Similar triangles have corresponding sides that are proportional, meaning their ratios are equal, but the sides themselves are not necessarily equal in length. A common misconception is to assume congruence from rigid motions alone, but including dilation accounts for possible size differences in similarity. To apply this in other problems, always check for matching angles first before examining side lengths or proportions.

Question 7

Two triangles overlap in the plane: $\triangle JKL$ and $\triangle MNL$ . Angle markings show $\angle J$ and $\angle M$ each have a single arc, and $\angle K$ and $\angle N$ each have a double arc. No side lengths are given, and there are no tick marks on any sides. The diagram is not drawn to scale.

Which relationship follows from the angle markings?

$\triangle JKL \sim \triangle MNL$ by AA, so $\dfrac{JK}{MN} = \dfrac{KL}{NL}$ . (correct answer)
$\triangle JKL \cong \triangle MNL$ because two angles are congruent.
$JK = MN$ because $\angle J \cong \angle M$ .
$\triangle JKL \sim \triangle MNL$ because $JL = ML$ is implied by the overlap.

Explanation: The AA criterion is a key method for proving triangle similarity. If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. In this diagram, angle J is marked congruent to angle M with a single arc, and angle K is marked congruent to angle N with a double arc. Therefore, by the AA similarity criterion, triangle JKL is similar to triangle MNL. Similar triangles have corresponding sides that are proportional, meaning their ratios are equal, but the sides themselves are not necessarily equal in length. A common misconception is to assume side equality from overlapping figures, but without measurements, proportionality is what follows from angle similarity. To apply this in other problems, always check for matching angles first before examining side lengths or proportions.

Question 8

In the diagram, triangles $\triangle TUV$ and $\triangle WXY$ are drawn in the plane. Angle markings show $\angle T \cong \angle W$ with one arc and $\angle U \cong \angle X$ with two arcs. No side lengths are labeled or marked, and the diagram is not drawn to scale. Which relationship follows from the angle markings?

$TU=WX$ because corresponding angles are congruent.
$\triangle TUV \cong \triangle WXY$ because two angles are marked congruent.
$\triangle TUV \sim \triangle WXY$ by AA, so $\dfrac{TV}{WY}=\dfrac{UV}{XY}$ . (correct answer)
$\triangle TUV \sim \triangle WYX$ by AA because $\angle T \cong \angle W$ and $\angle U \cong \angle Y$ .

Explanation: The skill being assessed is the AA criterion for triangle similarity. The AA criterion states that if two pairs of corresponding angles in two triangles are congruent, then the triangles are similar, as this permits a similarity transformation to map one onto the other. In this problem, the marked angles are angle T congruent to angle W with one arc and angle U congruent to angle X with two arcs. Applying the AA criterion, triangle TUV is similar to triangle WXY with correspondence T to W, U to X, and V to Y. Because the triangles are similar, their corresponding sides are proportional, meaning the ratios of the lengths of corresponding sides are equal, but the sides themselves are not necessarily equal in length. A common misconception is to misalign correspondences, such as linking angle U to angle Y instead of X, leading to incorrect similarity statements. When approaching similar problems, always check for matching angles first to establish similarity before comparing side lengths or ratios.

Question 9

Triangle PQR undergoes a composition of transformations: first a rotation of 90° counterclockwise about the origin, then a dilation with scale factor 2.5 about point (1, 1), producing triangle P'Q'R'. Given that $\angle P = 29°$ , $\angle Q = 108°$ , $\angle P' = 29°$ , and $\angle Q' = 108°$ , a student wants to use the AA criterion to prove similarity. What additional step is most important for a complete proof?

Calculate $\angle R$ and $\angle R'$ to verify that all three pairs of corresponding angles are equal, strengthening the similarity argument
Verify that the composition of transformations actually maps triangle PQR to triangle P'Q'R' by checking intermediate vertex positions
Confirm that angles P and P' are corresponding angles, and angles Q and Q' are corresponding angles, based on the transformation mapping (correct answer)
Measure the side lengths of both triangles to ensure that the sides are proportional with ratio 2.5, validating the dilation effect

Explanation: For the AA criterion to apply, the student must verify that the equal angles are actually corresponding angles between the triangles. Just because angles have the same measure and similar labels doesn't guarantee they correspond under the given transformation. The student needs to trace how the rotation and dilation map specific vertices to confirm angle correspondence. Choice A is unnecessary since two equal angle pairs suffice for AA criterion. Choice B focuses on mechanics rather than logical proof structure. Choice D is unnecessary since AA criterion alone establishes similarity and proportional sides.

Question 10

Triangle UVW is transformed by a dilation with scale factor $\frac{3}{2}$ centered at point P, followed by a reflection across line ℓ, to produce triangle U'V'W'. A student measures $\angle V = 71°$ in the original triangle and $\angle V' = 71°$ in the transformed triangle. The student then claims that if $\angle U = \angle U' = 44°$ , then the AA criterion confirms the triangles are similar. Which aspect of this reasoning needs clarification?

The reasoning is correct; the AA criterion is properly applied since two pairs of corresponding angles are verified as equal
The student should verify that the dilation center P and reflection line ℓ are positioned to maintain angle correspondence
The reasoning is flawed because reflections can change angle measures, contradicting the assumption that $\angle V' = 71°$
The student should confirm that $\angle U$ and $\angle U'$ are actually corresponding angles in their respective triangles (correct answer)

Explanation: While similarity transformations preserve angle measures, the student must verify that the angles being compared are actually corresponding angles between the two triangles. The labeling U, V, W and U', V', W' suggests correspondence, but this needs to be confirmed based on how the transformation maps the vertices. If the angles are indeed corresponding, then AA criterion applies and the triangles are similar. Choice A assumes correspondence without verification. Choice B incorrectly suggests that positioning affects angle preservation. Choice C incorrectly states that reflections change angle measures.

Question 11

In triangle PQR, $\angle P = 38°$ and $\angle Q = 67°$ . Triangle STU is formed by applying a dilation with center O and scale factor $\frac{2}{3}$ , followed by a 180° rotation about point O to triangle PQR. If $\angle S = 38°$ and $\angle T = 67°$ , what can be concluded about the similarity of the triangles?

The triangles are similar by AA criterion since two pairs of corresponding angles are equal, regardless of the transformation order (correct answer)
The triangles are similar only because the dilation was applied first; reversing the transformation order would change the angle measures
The triangles are not similar because the 180° rotation changes the orientation, which affects the angle-angle correspondence
The triangles are similar by AA criterion, but only if the scale factor is positive and the rotation preserves vertex labeling

Explanation: Similarity transformations (dilations, rotations, reflections, translations) always preserve angle measures regardless of their order of application. Since

\angle P = \angle S = 38°

and

\angle Q = \angle T = 67°

, we have two pairs of equal corresponding angles, satisfying the AA criterion for similarity. Choice B incorrectly suggests that transformation order affects angle measures. Choice C incorrectly claims that orientation changes affect similarity. Choice D incorrectly adds unnecessary conditions about scale factor sign and vertex labeling.

Question 12

Consider triangle XYZ with $\angle X = 35°$ and $\angle Y = 72°$ . A series of similarity transformations produces triangle X₁Y₁Z₁ where $\angle X₁ = 35°$ and $\angle Z₁ = 78°$ . When a student applies the AA criterion to determine if the triangles are similar, what issue will the student encounter?

The triangles satisfy AA criterion since $\angle X = \angle X₁$ and both triangles have the same third angle measure
The triangles do not satisfy AA criterion because $\angle Z$ in triangle XYZ equals 73°, not matching $\angle Z₁ = 78°$ (correct answer)
The triangles cannot be similar because similarity transformations should preserve all angle measures exactly
The student must verify that the angles being compared are actually corresponding angles between the triangles

Explanation: In triangle XYZ:

\angle Z = 180° - 35° - 72° = 73°

. In triangle X₁Y₁Z₁:

\angle Y₁ = 180° - 35° - 78° = 67°

. We have

\angle X = \angle X₁ = 35°

, but no other angle pairs match:

\angle Y = 72° ≠ 67° = \angle Y₁

and

\angle Z = 73° ≠ 78° = \angle Z₁

. This creates a contradiction because similarity transformations should preserve angle measures, suggesting an error in the given information or measurements. Choice A incorrectly assumes angle correspondence. Choice C correctly identifies the issue. Choice D, while generally good practice, doesn't address the fundamental angle mismatch.

Question 13

In the diagram, triangle $\triangle A'B'C'$ appears to be a transformed image of triangle $\triangle ABC$ . Angles $\angle A$ and $\angle A'$ are marked congruent with one arc, and angles $\angle B$ and $\angle B'$ are marked congruent with two arcs. No side lengths are shown, and the diagram is not drawn to scale. Which criterion justifies triangle similarity?

AA, because two pairs of corresponding angles are marked congruent. (correct answer)
AAA, because all three angles must be marked to claim similarity.
SSA, because two angles imply a matching side.
Congruence, because a transformation image must preserve side lengths.

Explanation: The skill being assessed is the AA criterion for triangle similarity. The AA criterion states that if two pairs of corresponding angles in two triangles are congruent, then the triangles are similar, as a similarity transformation can align the figures while preserving angle measures. In this problem, the marked angles are angle A congruent to angle A' with one arc and angle B congruent to angle B' with two arcs. Applying the AA criterion, triangle ABC is similar to triangle A'B'C' with correspondence A to A', B to B', and C to C'. Because the triangles are similar, their corresponding sides are proportional, meaning the ratios of the lengths of corresponding sides are equal, but the sides themselves are not necessarily equal in length. A common misconception is to require all three angles to be marked for similarity, but since the third angles are automatically congruent by the triangle angle sum, two suffice. When approaching similar problems, always check for matching angles first to establish similarity before comparing side lengths or ratios.

Question 14

Two triangles $\triangle GHI$ and $\triangle JKL$ are shown. Angles $\angle G$ and $\angle J$ are marked congruent with one arc, and angles $\angle H$ and $\angle K$ are marked congruent with two arcs. No side lengths are labeled, and the diagram is not drawn to scale. Which claim about side lengths must be true?

$\dfrac{GH}{JK}=\dfrac{HI}{KL}$ (correct answer)
$GH=JK$
$\dfrac{GH}{HI}=\dfrac{JK}{JL}$
$\dfrac{GI}{JL}=1$

\angle G

congruent to

\angle J

with one arc and

\angle H

congruent to

\angle K

with two arcs. Applying the AA criterion,

\triangle GHI

is similar to

\triangle JKL

with correspondence G to J, H to K, and I to L. Because the triangles are similar, their corresponding sides are proportional, meaning the ratios of the lengths of corresponding sides are equal, but the sides themselves are not necessarily equal in length. A common misconception is to equate non-corresponding side ratios, such as comparing

GH

HI

with

JK

JL

, which ignores the proper vertex mapping. When approaching similar problems, always check for matching angles first to establish similarity before comparing side lengths or ratios.

Question 15

Two triangles $\triangle EFG$ and $\triangle HIJ$ are shown in the plane. Angle markings indicate $\angle F \cong \angle I$ and $\angle G \cong \angle J$ . Which criterion justifies triangle similarity?

AA, because two pairs of corresponding angles are congruent. (correct answer)
SSA, because two angles determine a side proportion.
Congruence, because matching angles make the triangles identical.
Measurement, because the diagram shows the triangles have the same area.

Explanation: This question directly asks which criterion justifies triangle similarity given angle information. The AA (Angle-Angle) criterion states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. The problem states that ∠F ≅ ∠I and ∠G ≅ ∠J, which gives us exactly two pairs of congruent corresponding angles. This matches the requirements of the AA criterion perfectly, establishing that △EFG ∼ △HIJ. Choice B mentions SSA, which is not a valid similarity criterion and doesn't apply to angle-only information. Choice C incorrectly suggests congruence when angle information alone can only establish similarity. When identifying similarity criteria from given information, AA requires exactly two pairs of congruent angles and is the only criterion that uses angle information exclusively.

Question 16

Two triangles $\triangle A_1B_1C_1$ and $\triangle A_2B_2C_2$ are shown. The angle arcs indicate $\angle B_1 \cong \angle B_2$ and $\angle C_1 \cong \angle C_2$ . Which relationship follows from the angle markings?

$\triangle A_1B_1C_1 \sim \triangle A_2B_2C_2$ , so $\frac{A_1B_1}{A_2B_2}=\frac{B_1C_1}{B_2C_2}$ . (correct answer)
$\triangle A_1B_1C_1 \cong \triangle A_2B_2C_2$ because two angles match.
$A_1B_1=A_2B_2$ because corresponding angles are congruent.
$\triangle A_1B_1C_1 \sim \triangle A_2B_2C_2$ because $\angle B_1$ matches $\angle C_2$ and $\angle C_1$ matches $\angle B_2$ .

Explanation: This question tests understanding of the AA criterion for triangle similarity. The AA criterion states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. The angle markings show ∠B₁ ≅ ∠B₂ and ∠C₁ ≅ ∠C₂, providing two pairs of congruent corresponding angles. By the AA criterion, this proves △A₁B₁C₁ ∼ △A₂B₂C₂, which means the triangles have the same shape with proportional sides. Similar triangles have the property that ratios of corresponding sides are equal, so A₁B₁/A₂B₂ = B₁C₁/B₂C₂ = A₁C₁/A₂C₂. Choice B incorrectly claims congruence when we only have similarity, while choice C wrongly assumes equal sides from angle congruence. Choice D has incorrect angle correspondence. To apply AA correctly, match corresponding vertices and verify two angle pairs before concluding proportional sides.

Question 17

In the diagram, $\triangle ABC$ and $\triangle A'B'C'$ are drawn. Angle markings show $\angle A \cong \angle A'$ (single arc) and $\angle C \cong \angle C'$ (double arc). No side lengths are labeled, and no sides are marked congruent. The diagram is not drawn to scale.

Which criterion justifies triangle similarity?

AA, because two pairs of corresponding angles are marked congruent. (correct answer)
AAA, because all three angles are marked congruent.
SSS, because the triangles appear to have proportional sides.
ASA, because two angles and a side are marked.

Explanation: The AA criterion is a key method for proving triangle similarity. If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. In this diagram, angle A is marked congruent to angle A' with a single arc, and angle C is marked congruent to angle C' with a double arc. Therefore, by the AA similarity criterion, triangle ABC is similar to triangle A'B'C'. Similar triangles have corresponding sides that are proportional, meaning their ratios are equal, but the sides themselves are not necessarily equal in length. A common misconception is to use AAA as a criterion, but since the third angle is automatically equal, AA is sufficient for similarity. To apply this in other problems, always check for matching angles first before examining side lengths or proportions.

Question 18

Two triangles $\triangle RST$ and $\triangle XYZ$ are shown in the plane. Angle markings show $\angle R$ and $\angle X$ each have a single arc, and $\angle S$ and $\angle Y$ each have a double arc. No side lengths are given, and the diagram is not drawn to scale.

Which claim about side lengths must be true?

$RS = XY$ because $\angle R \cong \angle X$ and $\angle S \cong \angle Y$ .
$\dfrac{RS}{XY} = \dfrac{ST}{YZ}$ because the triangles are similar by AA. (correct answer)
$RT = XZ$ because the triangles have two equal angles.
$\dfrac{RS}{ST} = \dfrac{XZ}{YZ}$ because $T$ corresponds to $Z$ .

Explanation: The AA criterion is a key method for proving triangle similarity. If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. In this diagram, angle R is marked congruent to angle X with a single arc, and angle S is marked congruent to angle Y with a double arc. Therefore, by the AA similarity criterion, triangle RST is similar to triangle XYZ. Similar triangles have corresponding sides that are proportional, meaning their ratios are equal, but the sides themselves are not necessarily equal in length. A common misconception is to assume equal side lengths from matching angles alone, but proportionality is the correct relationship without size information. To apply this in other problems, always check for matching angles first before examining side lengths or proportions.

Question 19

A triangle $\triangle GHI$ is shown, and another triangle $\triangle JKL$ is shown elsewhere in the plane. Angle markings show $\angle G$ and $\angle J$ each have a single arc, and $\angle H$ and $\angle K$ each have a double arc. No side lengths are labeled, and the diagram is not drawn to scale.

Which conclusion about the triangles is valid?

$\triangle GHI \sim \triangle JKL$ by AA, so $\dfrac{GH}{JK} = \dfrac{HI}{KL}$ . (correct answer)
$\triangle GHI \cong \triangle JKL$ by AA.
$GH = JK$ and $HI = KL$ because the angles are marked congruent.
The triangles are similar only if $GI = JL$ is given.

Explanation: The AA criterion is a key method for proving triangle similarity. If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. In this diagram, angle G is marked congruent to angle J with a single arc, and angle H is marked congruent to angle K with a double arc. Therefore, by the AA similarity criterion, triangle GHI is similar to triangle JKL. Similar triangles have corresponding sides that are proportional, meaning their ratios are equal, but the sides themselves are not necessarily equal in length. A common misconception is to require a matching side for similarity, but AA alone establishes the similarity without side information. To apply this in other problems, always check for matching angles first before examining side lengths or proportions.

Question 20

A triangle $\triangle MNO$ is drawn, and a second triangle $\triangle PQR$ is drawn in the plane. Angle markings show $\angle M$ and $\angle P$ each have a single arc, and $\angle N$ and $\angle Q$ each have a double arc. No side lengths are labeled, and the diagram is not drawn to scale.

Which relationship follows from the angle markings?

$MN:NO = PQ:QR$ because the triangles are similar by AA. (correct answer)
$MN = PQ$ because the triangles have two equal angles.
$\triangle MNO$ is congruent to $\triangle PQR$ because AA holds.
$MN:MO = PQ:PR$ because $O$ corresponds to $Q$ .

Explanation: The AA criterion is a key method for proving triangle similarity. If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. In this diagram, angle M is marked congruent to angle P with a single arc, and angle N is marked congruent to angle Q with a double arc. Therefore, by the AA similarity criterion, triangle MNO is similar to triangle PQR. Similar triangles have corresponding sides that are proportional, meaning their ratios are equal, but the sides themselves are not necessarily equal in length. A common misconception is to set up proportions incorrectly by mismatching corresponding vertices, but proper angle correspondence ensures accurate ratios. To apply this in other problems, always check for matching angles first before examining side lengths or proportions.