Geometry Quiz: Triangle Congruence From Rigid Motions

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Triangle Congruence From Rigid MotionsQuestion 1 of 9

Triangle $PQR$ with vertices $P(3, 4)$ , $Q(7, 2)$ , $R(5, 8)$ is congruent to triangle $STU$ . If the congruence can be established through rigid motions, but triangle $STU$ has vertices $S(-4, 3)$ , $T(-2, 7)$ , $U(-8, 5)$ , what must be verified to confirm the triangles are indeed congruent?

Check that corresponding angles are equal and that no dilations were used in the transformation

Confirm that the triangles have the same perimeter and the same area measurements

Verify that one triangle can be mapped onto the other using only translations and rotations

Calculate the side lengths of both triangles and verify they form the same set of distances

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Geometry Quiz

Geometry Quiz: Triangle Congruence From Rigid Motions

Practice Triangle Congruence From Rigid Motions 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 Triangle Congruence From Rigid Motions, giving you a quick way to practice the rules, question types, and explanations that matter most for Geometry.

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

Check that corresponding angles are equal and that no dilations were used in the transformation
Confirm that the triangles have the same perimeter and the same area measurements
Verify that one triangle can be mapped onto the other using only translations and rotations
Calculate the side lengths of both triangles and verify they form the same set of distances (correct answer)

Explanation: When determining congruence between triangles using coordinate geometry, you need to verify that corresponding sides have identical lengths. Congruent triangles have exactly the same shape and size, which means all corresponding sides must be equal. For triangle

PQR

, calculate the side lengths using the distance formula

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

$PQ = \sqrt{(7-3)^2 + (2-4)^2} = \sqrt{16+4} = \sqrt{20}$
$QR = \sqrt{(5-7)^2 + (8-2)^2} = \sqrt{4+36} = \sqrt{40}$
$PR = \sqrt{(5-3)^2 + (8-4)^2} = \sqrt{4+16} = \sqrt{20}$

For triangle

STU

$ST = \sqrt{(-2-(-4))^2 + (7-3)^2} = \sqrt{4+16} = \sqrt{20}$
$TU = \sqrt{(-8-(-2))^2 + (5-7)^2} = \sqrt{36+4} = \sqrt{40}$
$SU = \sqrt{(-8-(-4))^2 + (5-3)^2} = \sqrt{16+4} = \sqrt{20}$

Both triangles have the same set of side lengths:

\{\sqrt{20}, \sqrt{20}, \sqrt{40}\}

, confirming congruence by SSS. Option A is incorrect because checking angles isn't necessary when you can prove SSS congruence, and mentioning dilations is irrelevant. Option B is wrong because equal perimeter and area don't guarantee congruence—different triangles can share these measurements. Option C is incomplete because it doesn't specify what needs verification about the mapping. Strategy tip: For coordinate geometry congruence problems, always calculate and compare corresponding side lengths first. The SSS congruence test is the most direct method when working with coordinates.

Question 2

Triangle $RST$ is congruent to triangle $UVW$ with the correspondence $R \leftrightarrow U$ , $S \leftrightarrow V$ , $T \leftrightarrow W$ . If a $270°$ counterclockwise rotation about point $P$ maps triangle $RST$ onto triangle $UVW$ , which statement about the angle measures is necessarily true?

Each angle in triangle $UVW$ is $270°$ greater than the corresponding angle in triangle $RST$
The sum of corresponding angles from both triangles equals $360°$ for each pair
$\angle R = \angle U$ , $\angle S = \angle V$ , and $\angle T = \angle W$ regardless of the rotation (correct answer)
The angles in triangle $UVW$ are the supplements of the corresponding angles in triangle $RST$

Explanation: Rigid motions, including rotations, preserve angle measures. Therefore, corresponding angles are congruent regardless of the specific rotation applied. The 270° rotation affects the position and orientation of the triangle, not the individual angle measures within it. Choices A, B, and D incorrectly suggest that the rotation angle affects the triangle's internal angle measures.

Question 3

Triangle $JKL$ undergoes a rigid motion to produce triangle $MNO$ . If $JK = 15$ , $KL = 9$ , $JL = 12$ , and $MN = 9$ , $NO = 12$ , $OM = 15$ , which additional information is needed to conclude that the triangles are congruent?

The correspondence between vertices must be established to verify angle congruence (correct answer)
At least one pair of corresponding angles must be measured and verified equal
The specific type of rigid motion (translation, rotation, or reflection) must be identified
No additional information is needed since corresponding sides are already congruent

Explanation: While all corresponding sides are equal in length (SSS), we need to establish which vertices correspond to conclude the triangles are congruent. The given side lengths could correspond in multiple ways (J↔M, K↔N, L↔O or other arrangements). Choice B is unnecessary since SSS guarantees angle congruence. Choice C is irrelevant for proving congruence. Choice D ignores the correspondence issue.

Question 4

Triangle $ABC$ is congruent to triangle $DEF$ by a sequence of rigid motions. Given that $AB = 13$ cm, $BC = 8$ cm, $AC = 15$ cm, and in triangle $DEF$ : $DE = 8$ cm, $EF = 15$ cm, $DF = 13$ cm, which correspondence between vertices ensures the congruence?

$A \leftrightarrow D$ , $B \leftrightarrow E$ , $C \leftrightarrow F$ with all sides matching in order
$A \leftrightarrow F$ , $B \leftrightarrow D$ , $C \leftrightarrow E$ based on matching side lengths (correct answer)
$A \leftrightarrow E$ , $B \leftrightarrow F$ , $C \leftrightarrow D$ to align equal sides properly
Any correspondence works since all sides are different lengths in each triangle

Explanation: We must match corresponding sides: AB = 13 matches DF = 13, so A↔F and B↔D. BC = 8 matches DE = 8, so B↔D and C↔E. AC = 15 matches EF = 15, so A↔F and C↔E. This gives correspondence A↔F, B↔D, C↔E. Choice A incorrectly matches sides in order. Choice C creates wrong pairings. Choice D ignores the need for proper correspondence.

Question 5

In triangle $DEF$ , $DE = 7$ cm, $EF = 10$ cm, and $\angle E = 65°$ . Triangle $GHI$ has $GH = 10$ cm, $HI = 7$ cm, and $\angle H = 65°$ . A student claims these triangles are congruent by SAS and can be mapped onto each other by rigid motions. Which analysis of this claim is correct?

The claim is correct because both triangles have two equal sides and an equal included angle
The claim is incorrect because the sides are not in the same order relative to the given angle
The claim is correct because SAS congruence guarantees the existence of a rigid motion mapping (correct answer)
The claim is incorrect because rigid motions require SSS congruence, not SAS congruence

Explanation: The triangles satisfy SAS congruence: DE = HI = 7 cm, EF = GH = 10 cm, and ∠E = ∠H = 65°. When two triangles are congruent by any valid method (SAS, SSS, ASA, AAS), there exists a sequence of rigid motions that maps one onto the other. Choice A misses the correspondence verification. Choice B incorrectly suggests the sides aren't properly arranged. Choice D incorrectly states that only SSS works with rigid motions.

Question 6

Two triangles are shown in the coordinate plane. Triangle $ABC$ has been transformed by exactly one rigid motion to produce triangle $A'B'C'$ . Based on the positions shown, which conclusion about the congruence of these triangles is most justified?

The triangles are congruent because they appear to have the same shape and size
The triangles are congruent because rigid motions always preserve distance and angle measures (correct answer)
The triangles may not be congruent because the transformation could include a dilation
Congruence cannot be determined without measuring all corresponding sides and angles

Explanation: Since triangle A'B'C' was produced by exactly one rigid motion applied to triangle ABC, and rigid motions (translations, rotations, reflections) preserve all distances and angle measures by definition, the triangles must be congruent. Choice A relies on visual approximation. Choice C incorrectly suggests dilation is a rigid motion. Choice D ignores the given information about rigid motion.

Question 7

The figure shows triangle $ABC$ and triangle $XYZ$ . If these triangles are congruent and triangle $ABC$ can be mapped onto triangle $XYZ$ by a reflection across line $\ell$ followed by a translation, which property is preserved throughout this transformation?

The orientation of the triangle remains the same after both transformations
All side lengths and angle measures remain unchanged after both transformations (correct answer)
The triangle's position relative to line $\ell$ stays constant throughout the process
The area increases after reflection but returns to original size after translation

Explanation: Both reflection and translation are rigid motions, which preserve all distances and angle measures. Choice A is incorrect because reflection changes orientation (though translation preserves it). Choice C is wrong because translation moves the triangle relative to line ‚Ñì. Choice D is incorrect because rigid motions preserve area throughout - it never changes.

Question 8

Two triangles are positioned as shown in the coordinate plane. Triangle $MNO$ appears to be the image of triangle $JKL$ after a sequence of rigid motions. If $JK = 6$ units and $\angle J = 42¬∞$ , which statement about triangle $MNO$ can be concluded with certainty?

Side $MN = 6$ units if $J$ corresponds to $M$ and $K$ corresponds to $N$
The triangle has an angle measuring $42¬∞$ , but its location depends on vertex correspondence (correct answer)
$\angle M = 42¬∞$ and the triangle's area equals that of triangle $JKL$
All sides of triangle $MNO$ are 6 units long since rigid motions preserve all measurements

Explanation: Since rigid motions preserve angle measures, triangle MNO must have an angle of 42¬∞, but we cannot determine which angle without knowing the correspondence between vertices. Choice A assumes a specific correspondence. Choice C assumes M corresponds to J without justification. Choice D incorrectly assumes all sides of triangle JKL are 6 units when only JK is given as 6 units.

Question 9

In the figure, triangle $PQR$ is mapped onto triangle $STU$ by a sequence of rigid motions. Given that $\angle P = 47¬∞$ , $PQ = 8$ cm, and $QR = 12$ cm, which statement about triangle $STU$ must be true?

$\angle S = 47¬∞$ and the triangle has a perimeter greater than 20 cm
$\angle T = 47¬∞$ and side $ST = 8$ cm, assuming $P$ corresponds to $T$
The triangle has the same area as triangle $PQR$ but different angle measures
$\angle S = 47¬∞$ if $P$ corresponds to $S$ , and all side lengths are preserved (correct answer)

Explanation: Since rigid motions preserve all distances and angle measures, corresponding angles and sides are congruent. If P corresponds to S, then ‚à†S = ‚à†P = 47¬∞, and all side lengths are preserved. Choice A doesn't specify correspondence. Choice B assumes wrong correspondence without justification. Choice C is incorrect because rigid motions preserve both area AND angle measures.