If corresponding sides are parallel after one transformation, the transformation preserved the orientation and direction of the sides, which means it was either a translation or rotation (not a reflection). Since the hexagons aren't aligned yet, additional transformations may be needed. Choice A is too restrictive about the type and number. Choice B incorrectly limits to one additional transformation. Choice D incorrectly identifies the first transformation as a reflection.

While both pentagons have the same rotational symmetry (72° rotations), this internal symmetry doesn't determine how to map one pentagon to another. The pentagons are in different locations and possibly different orientations, so the mapping requires transformations to change position and/or orientation. A rotation about the center of one pentagon cannot move it to the location of the other pentagon. Choice A incorrectly assumes rotational symmetry determines the mapping. Choice B incorrectly suggests any rotation works. Choice D ignores the need to change position.

Since the right angles point in opposite directions while the hypotenuses are parallel, the triangles have opposite orientations, so a reflection is needed. Since they're in different positions, a translation is also needed. These two transformations are sufficient and necessary. Choice A suggests a single reflection could work, but the specific reflection described wouldn't necessarily map the triangles correctly. Choice C could work but isn't minimal since the 180° rotation would need to be about a very specific point. Choice D uses too many transformations.

For a single reflection to map triangle ABC to triangle DEF, the line of reflection must be the perpendicular bisector of the segment connecting any pair of corresponding vertices. Since it's a reflection, this perpendicular bisector property will automatically apply to all corresponding vertex pairs. Choice A incorrectly specifies requirements about the centroid. Choice C incorrectly suggests the line should be parallel to a side. Choice D incorrectly involves the circumcenter and longest side, which aren't relevant to reflection mapping.

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.