This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via such transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other; for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation preserving size and shape, the triangles are congruent; if they had different sizes like sides 3-4-5 vs 6-8-10, they would not be congruent as that would require a dilation, which changes size. For ABC at (2,3),(6,3),(4,6) to DEF at (-2,-3),(-6,-3),(-4,-6), applying 180° rotation about origin negates both coordinates: (2,3) to (-2,-3), etc., mapping exactly. Therefore, choice A is correct, confirming congruence via rotation. A common error is selecting translation like (-4,-6) (choice C), which would map (2,3) to (-2,-3) but (6,3) to (2,-3) not matching. To find the sequence: (1) compare figures (same size: base 4, etc.), (2) identify 180° turn, (3) build as rotation about origin, (4) verify all points match, (5) simplest single transformation. Congruence means same size and shape via rigid transformations only, no scaling; common errors include incomplete sequences or wrong center of rotation.

This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via these transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation that preserves size and shape, the triangles are congruent. If the figures had different sizes, such as sides 3-4-5 versus 6-8-10, they would not be congruent because that would require a dilation (scaling by 2), which isn't a rigid transformation as it changes the size. The triangles map by negating x-coordinates while keeping y: R(-2,1) to (2,1), S(-5,1) to (5,1), T(-3,4) to (3,4), which is a reflection over the y-axis. So, choice B is correct, showing congruence through this rigid motion. A common error is choosing rotation or translation that doesn't align points or claiming size change. To find the sequence: (1) compare (same size), (2) identify flip over y-axis, (3) build reflection, (4) verify matches, (5) simplest; rigid transformations preserve size, unlike dilations for similarity.

This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via these transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation that preserves size and shape, the triangles are congruent. If the figures had different sizes, such as sides 3-4-5 versus 6-8-10, they would not be congruent because that would require a dilation (scaling by 2), which isn't a rigid transformation as it changes the size. For the pentagon, each point's y-coordinate is negated while x remains: A(1,1) to (1,-1), B(3,1) to (3,-1), up to E(0,2) to (0,-2), a reflection over the x-axis mapping exactly step-by-step. Therefore, choice A is correct, confirming congruence via this rigid transformation. A common error is using wrong transformation like rotation that doesn't fit or including dilation unnecessarily. To find the sequence: (1) compare (same size), (2) identify flip over x-axis, (3) build reflection, (4) verify all vertices, (5) simplest; congruence excludes non-rigid changes like scaling.

This question tests understanding of congruent figures obtainable from each other by a sequence of rigid transformations (rotations, reflections, translations)—same size and shape means congruence via transformations. Two figures are congruent if a rigid transformation sequence maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—translation is a rigid transformation preserving size/shape, so the triangles are congruent; if different sizes (sides 3-4-5 vs 6-8-10), not congruent—would need dilation (scaling 2×) which isn't a rigid transformation (changes size); sequence description: identify transformations needed (flip? turn? shift?), order them (reflect first then translate, or rotate then reflect), verify maps all vertices correctly. For polygon JKLM with J(2,2), K(5,2), L(5,5), M(2,5) and J'K'L'M' with J'(-2,-2), K'(-5,-2), L'(-5,-5), M'(-2,-5), rotating 180° about origin maps (x,y) to (-x,-y): J to (-2,-2), K to (-5,-2), L to (-5,-5), M to (-2,-5), matching exactly. This 180° rotation is the correct rigid transformation that maps JKLM to J'K'L'M', confirming congruence. A common error is choosing a translation like (-4,-4), which would map J to (-2,-2) but K to (1,-2), not matching. To find the sequence: (1) compare figures (same size? check side lengths, angles), (2) identify orientation difference (flipped? rotated? just shifted?), (3) build sequence (if flipped: reflection needed, if rotated: rotation needed, if different position: translation), (4) verify (apply transformations to figure 1, should get figure 2 exactly—all vertices match), (5) simplify if possible (fewest transformations needed). Congruence means same size and shape, obtainable by rigid transformations only (rotation/reflection/translation), no scaling/stretching/skewing; common errors include including dilation (that's similarity), wrong order giving wrong final position, incomplete sequence (missing a needed transformation), or claiming congruence when sizes differ (not checking all measurements).

This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via such transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other; for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation preserving size and shape, the triangles are congruent; if they had different sizes like sides 3-4-5 vs 6-8-10, they would not be congruent as that would require a dilation, which changes size. Here, quadrilateral PQRS at P(1,1), Q(4,1), R(4,3), S(1,3) maps to P'(-1,1), Q'(-4,1), R'(-4,3), S'(-1,3) by negating x-coordinates while keeping y the same, which is a reflection over the y-axis applied step-by-step to each vertex. Therefore, the correct transformation is reflection over the y-axis, choice C, confirming congruence. A common error is choosing rotation like 180° about origin (choice B), which would map (1,1) to (-1,-1) not matching, or ignoring the flip in orientation. To find the sequence: (1) compare figures (same size: sides like PQ=3, P'Q'=3), (2) identify flipped over vertical axis, (3) build as reflection over y-axis, (4) verify applying to all points matches exactly, (5) simplest single transformation. Congruence means same size and shape via rigid transformations only, no scaling; common errors include using dilation for size changes or incorrect order leading to mismatch.

This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via these transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation that preserves size and shape, the triangles are congruent. If the figures had different sizes, such as sides 3-4-5 versus 6-8-10, they would not be congruent because that would require a dilation (scaling by 2), which isn't a rigid transformation as it changes the size. Subtracting 2 from both x and y maps ABC to A'B'C': (2,2) to (0,0), (6,2) to (4,0), (4,5) to (2,3), a translation by (-2,-2). Thus, choice A is correct, confirming the single rigid transformation for congruence. A common error is using reflection instead or incomplete sequence missing a step. To find the sequence: (1) compare (same size), (2) identify shift, (3) build translation (-2,-2), (4) verify all points, (5) simplest; congruence via rigid motions only, no skewing.

This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via such transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other; for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation preserving size and shape, the triangles are congruent; if they had different sizes like sides 3-4-5 vs 6-8-10, they would not be congruent as that would require a dilation, which changes size. For ABC at (-2,1),(-5,1),(-3,4) and A'B'C' at (2,1),(5,1),(3,4), negating x-coordinates (from negative to positive) while keeping y is a reflection over the y-axis, mapping each point directly. Therefore, choice C is correct, confirming congruence with the flip. A common error is choosing reflection over x-axis (choice B), which would negate y instead, not matching. To find the sequence: (1) compare figures (same size: base 3, etc.), (2) identify horizontal flip, (3) build as reflection over y-axis, (4) verify maps all vertices exactly, (5) simplest single step. Congruence means same size and shape via rigid transformations only, no scaling; common errors include selecting wrong axis for reflection or adding unnecessary translations.

This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via such transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other; for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation preserving size and shape, the triangles are congruent; if they had different sizes like sides 3-4-5 vs 6-8-10, they would not be congruent as that would require a dilation, which changes size. Here, GHI at (0,0),(3,0),(0,2) has sides 3,2,sqrt(13), while JKL at (0,0),(6,0),(0,4) has sides 6,4,sqrt(52)=2sqrt(13), showing a scale factor of 2, so only dilation maps them, not rigid transformations. Thus, they are not congruent, and the correct choice is C, as dilation is not rigid. A common error is claiming congruence via rotation and translation (choice B) without checking sizes differ. To find the sequence: (1) compare figures (sizes differ: bases 3 vs 6), (2) identify scaling, not just shift or turn, (3) no rigid sequence possible, (4) verification shows no exact match without scaling, (5) cannot simplify. Congruence means same size and shape via rigid transformations only, no scaling; common errors include ignoring size checks or confusing similarity with congruence.

This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via these transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation that preserves size and shape, the triangles are congruent. If the figures had different sizes, such as sides 3-4-5 versus 6-8-10, they would not be congruent because that would require a dilation (scaling by 2), which isn't a rigid transformation as it changes the size. The student's claim involves a 180° rotation about the origin, which sends (x,y) to (-x,-y): applying to J(1,2) gives (-1,-2), K(3,2) to (-3,-2), L(3,5) to (-3,-5), M(1,5) to (-1,-5), exactly matching the given points step-by-step. Therefore, the student is correct, as in choice A, since this rigid transformation maps the polygons congruently. A common error is thinking rotations change size or using incorrect mapping like (x,-y), which wouldn't align. To find the sequence: (1) compare sizes (match), (2) identify 180° turn, (3) build as rotation, (4) verify all points, (5) simplest; congruence excludes scaling, focuses on rigid motions.

This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via such transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other; for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation preserving size and shape, the triangles are congruent; if they had different sizes like sides 3-4-5 vs 6-8-10, they would not be congruent as that would require a dilation, which changes size. For triangle ABC at A(2,1), B(5,1), C(3,4) and A'(2,-1), B'(5,-1), C'(3,-4), negating y-coordinates while keeping x the same is a reflection over the x-axis, mapping each vertex accordingly in one step. Thus, the correct choice is B, reflection over the x-axis, proving congruence with reversed orientation. A common error is selecting translation like (0,-2) (choice C), which shifts but doesn't flip, resulting in mismatch like (2,-1) but others not aligning fully. To find the sequence: (1) compare figures (same size: AB=3, A'B'=3), (2) identify flipped over horizontal axis, (3) build as reflection over x-axis, (4) verify application matches all vertices, (5) simplest as single transformation. Congruence means same size and shape via rigid transformations only, no scaling; common errors include claiming non-congruence for orientation differences or using rotation instead of reflection.