Triangle Congruence from Rigid Motions - Geometry

1. 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?
   Answer: $\angle R = \angle U$, $\angle S = \angle V$, and $\angle T = \angle W$ regardless of the rotation 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.
2. 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?
   Answer: The triangles are congruent because rigid motions always preserve distance and angle measures 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.
3. 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?
   Answer: All side lengths and angle measures remain unchanged after both transformations 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.
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?
   Answer: $A \leftrightarrow F$, $B \leftrightarrow D$, $C \leftrightarrow E$ based on matching side lengths 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.
5. 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?
   Answer: $\angle S = 47°$ if $P$ corresponds to $S$, and all side lengths are preserved Since rigid motions preserve all distances and angle measures, corresponding angles and sides are congruent. If P corresponds to S, then ∠S = ∠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.
6. 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?
   Answer: The triangle has an angle measuring $42°$, but its location depends on vertex correspondence 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.
7. 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?
   Answer: The correspondence between vertices must be established to verify angle congruence 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.
8. 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?
   Answer: Calculate the side lengths of both triangles and verify they form the same set of distances 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.
9. 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?
   Answer: The claim is correct because SAS congruence guarantees the existence of a rigid motion mapping 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.

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