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.

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 polygons WXYZ at (0,1),(2,1),(2,4),(0,4) and W'X'Y'Z' at (3,2),(5,2),(5,5),(3,5), each point shifts by (3,1), mapping W to W', X to X', etc., exactly. Therefore, they are congruent via translation by (3,1), choice B. A common error is claiming not congruent due to position difference (choice A), ignoring translations preserve congruence. To find the sequence: (1) compare figures (same size: width 2, height 3), (2) identify just a shift, no flip or rotation, (3) build as translation (3,1), (4) verify applies to all vertices matching, (5) simplest single transformation. Congruence means same size and shape via rigid transformations only, no scaling; common errors include thinking rotations change congruence or not checking 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. In this problem, comparing vertices shows A(1,2) to D(1,7) shifts by (0,5), B(4,2) to E(4,7) by (0,5), and C(2,5) to F(2,10) by (0,5), so a single translation maps them step-by-step. Thus, the correct choice is A, translation by (0,5), as it precisely overlays ABC onto DEF. A common error is selecting a reflection like choice B when no flip is needed, or confusing with non-rigid transformations if sizes differed. To find the sequence: (1) compare figures (same size: side lengths match, e.g., AB=3, DE=3), (2) identify just a vertical shift, no flip or rotation, (3) build sequence as translation (0,5), (4) verify by applying to all vertices, which match perfectly, (5) this is the simplest with one transformation. Congruence means same size and shape via rigid transformations only, no scaling; common errors include claiming congruence for similar but scaled figures or using wrong transformations that don't align points.

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 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, first reflecting ABC at (1,0),(3,0),(2,2) over y-axis gives (-1,0),(-3,0),(-2,2), then translating by (0,4) yields (-1,4),(-3,4),(-2,6), matching A'B'C' step-by-step. Thus, choice B is the correct sequence, proving congruence. A common error is reversing order like choice A, which translates first then reflects, resulting in different positions like ( -3,4) not matching. To find the sequence: (1) compare figures (same size: base 2, etc.), (2) identify flip and shift, (3) build as reflection then translation, (4) verify sequential application matches all, (5) this order is necessary. Congruence means same size and shape via rigid transformations only, no scaling; common errors include wrong sequence order or omitting the reflection.

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 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 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 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 triangle ABC with A(0,0), B(4,0), C(0,3) and DEF with D(0,0), E(8,0), F(0,6), the side lengths of ABC are AB=4, AC=3, BC=5, while DEF has DE=8, DF=6, EF=10, which is twice as large, so a dilation by factor 2 is needed, not just rigid transformations. Thus, the triangles are not congruent, as they have different sizes. A common error is claiming congruence via rotation when sizes differ, like option D which incorrectly states rotations are not allowed—they are, but size mismatch prevents congruence. 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 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 triangle ABC with A(1,2), B(4,2), C(2,5) and DEF with D(-1,2), E(-4,2), F(-2,5), reflecting over y-axis: x to -x, A to (-1,2), B to (-4,2), C to (-2,5), matching DEF exactly. Thus, the triangles are congruent via this reflection, as stated in option A. A common error is claiming non-congruence due to dilation when sizes are actually the same, or thinking reflections change lengths, which they don't. 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).