In a regular hexagon with consecutive vertices A, B, C, D, E, F, vertex A is directly opposite to vertex D, and vertex B is directly opposite to vertex E. A 180° rotation about the center maps each vertex to its opposite vertex, so A→D and B→E. Choice A is incorrect because a reflection across a line perpendicular to AB would not map A to D. Choice B is incorrect because a 120° clockwise rotation would map A to F, not D. Choice D is incorrect because a reflection across line AD would map A to D but would not map B to E.

A rectangle has exactly two lines of symmetry: one line through the center parallel to the longer sides (horizontal line through center), and one line through the center parallel to the shorter sides (vertical line through center). Choice A is incorrect because it only counts one of the two lines. Choice C is incorrect because diagonals are not lines of symmetry for a rectangle (unless it's a square). Choice D is incorrect for the same reason - rectangles do not have diagonal symmetry lines.

A parallelogram that is not a rectangle has 180° rotational symmetry about its center (point M), but no lines of reflection symmetry. The 180° rotation maps each vertex to the opposite vertex across the center. Choice A is incorrect because non-rectangular parallelograms have no reflection symmetries. Choice C is incorrect because the diagonals are not lines of symmetry unless the parallelogram is a rectangle. Choice D is incorrect because while there is one rotational symmetry (180°), there are no reflection symmetries.

When analyzing transformations that carry a figure onto itself, you're looking for the symmetries of that shape. A rhombus has specific symmetry properties that determine which transformations preserve it.

First, let's find where the diagonals intersect. Diagonal $$DF$$ connects $$(0,0)$$ to $$(8,4)$$, and diagonal $$EG$$ connects $$(3,4)$$ to $$(5,0)$$. The intersection point $$H$$ is at $$(4,2)$$. In any rhombus, the diagonals bisect each other at right angles, creating natural lines of reflection symmetry.

The correct answer is D because a rhombus has exactly two lines of reflection symmetry: the lines containing its diagonals. When you reflect the rhombus across diagonal $$DF$$ or diagonal $$EG$$, each vertex maps to another vertex of the rhombus, carrying the figure onto itself.

Choice A combines two reflections that don't correspond to the rhombus's natural symmetries. The lines $$y = 2$$ and $$x = 4$$ pass through point $$H$$ but aren't the diagonal lines.

Choice B is partially correct—a $$180°$$ rotation about $$H$$ does map the rhombus onto itself—but it's incomplete since reflection symmetries also work.

Choice C incorrectly assumes the rhombus has four-fold rotational symmetry. Only squares have $$90°$$ rotational symmetry; general rhombuses only have $$180°$$ rotational symmetry.

Strategy tip: Remember that rhombuses have exactly three types of symmetries: two diagonal reflections and one $$180°$$ rotation about the center. Don't confuse rhombus symmetries with square symmetries.

A regular $$n$$-gon has exactly $$n$$ rotational symmetries and $$n$$ lines of reflection symmetry. If polygon $$P$$ has 6 rotational symmetries, it must be a regular hexagon (6 sides). If polygon $$Q$$ has 3 reflection symmetries, it must be a regular triangle (3 sides). Since the problem states both polygons have the same number of sides, this creates a contradiction. Choice A is incorrect because a hexagon has 6, not 3, reflection lines. Choice B correctly identifies the polygons but contradicts the given condition. Choice C is incorrect because orientation doesn't change the number of symmetries.

In a regular octagon with vertices labeled consecutively P, Q, R, S, T, U, V, W, the rotation that maps P to T is a rotation by 3 positions (P→Q→R→S→T), which corresponds to 3 × (360°/8) = 135°. This same rotation maps every vertex 3 positions forward: Q maps to U. Choice B is incorrect because V would be 4 positions from Q. Choice C is incorrect because R is only 1 position from Q. Choice D is incorrect because W would be 6 positions from Q in the opposite direction.

When analyzing symmetries of geometric figures, you need to systematically consider all possible rotations and reflections, then determine which ones produce the specific mapping described.

A square has 8 total symmetries: 4 rotations ($$0°$$, $$90°$$, $$180°$$, $$270°$$) and 4 reflections (across two diagonals and two perpendicular bisectors of opposite sides). To find which symmetries map vertex $$J$$ to vertex $$L$$, visualize or sketch the square with vertices labeled consecutively.

Since $$J$$ and $$L$$ are diagonally opposite vertices, only two transformations can achieve this mapping. A $$180°$$ rotation about the center swaps each vertex with its diagonal opposite, sending $$J$$ to $$L$$. Additionally, reflection across the diagonal that doesn't contain $$J$$ or $$L$$ will also map $$J$$ to $$L$$ by "flipping" the square across that line.

Choice B is incorrect because it identifies only the rotational symmetry while missing the reflectional one. Choice C incorrectly includes $$90°$$ and $$270°$$ rotations, which would map $$J$$ to adjacent vertices $$K$$ or $$M$$, not to the diagonal $$L$$. Choice D vastly overcounts—there aren't four different symmetries that accomplish this specific mapping, and three different reflections certainly don't all send $$J$$ to $$L$$.

The correct answer is A: exactly two symmetries work.

Study tip: For symmetry problems, always sketch the figure with labeled vertices and systematically test each transformation type. Diagonal mappings in squares typically involve either $$180°$$ rotation or reflection across the "other" diagonal.

In a regular pentagon, the vertices are evenly spaced around the center. Each adjacent pair of vertices is separated by 360°/5 = 72°. To map vertex A to vertex C, we need to move 2 positions (A→B→C), so the rotation angle is 2 × 72° = 144°. Choice A gives the angle between adjacent vertices, not the angle to map A to C. Choice B gives the interior angle of the pentagon, which is irrelevant to rotational symmetry. Choice D gives a larger angle that would also work (moving 3 positions counterclockwise), but the question asks for the smallest positive angle.

This question asks about the symmetries of a regular octagon. A symmetry is a transformation that maps the polygon onto itself. Regular octagons have extensive symmetry properties due to their 8 equal sides and angles. The octagon has rotational symmetry of order 8, meaning it maps onto itself under rotations of 45°, 90°, 135°, 180°, 225°, 270°, and 315° about its center. Additionally, it has exactly 8 lines of reflection symmetry: 4 lines connecting opposite vertices and 4 lines connecting midpoints of opposite sides. These symmetries make the regular octagon one of the most symmetric polygons. Students might think it has only 4 lines (like a square) or forget to count all rotational positions. To find all symmetries of regular polygons, remember that an n-sided regular polygon has n rotational symmetries and n reflection lines.