Symmetry: slides, turns and flips

How many lines of symmetry does a square have?

A rectangle is 6 cm long and 3 cm wide. How many lines of symmetry does it have?

Which shape has rotational symmetry (order 2 or more) but no line of symmetry?

Which of these shapes have exactly 2 lines of symmetry? Choose from: square, rectangle, rhombus, kite, parallelogram, equilateral triangle. (Here the rectangle and the rhombus are not squares.) Write your answer like {kite, square}.

A shape is given a quarter turn clockwise about the point P. Which single move puts it back where it started?

A flag is flipped in a vertical mirror line, and then flipped in a second vertical mirror line 3 squares to the right of the first. The result is a slide. How many squares does the flag slide?

Amir says: "A flip in a vertical mirror line and a half turn always do the same thing." Which example shows that he is wrong?

Two mirror lines cross at an angle of 60°. A shape is flipped in the first mirror and then in the second. The result is a turn about the crossing point. Through how many degrees is it turned?

Lean proved flips_in_parallel_lines: flipAt d (flipAt c p) = slide (2 * (d - c)) 0 p for every point p. Here flipAt c flips in the vertical line x = c, and slide a b slides a squares right and b squares up. What single move is "flip in the line x = 1, then flip in the line x = 4"?

Points on squared paper are pairs (x, y). Complete flip(p) (flip in the vertical line through (0, 0), so left and right swap) and quarter_turn(p) (a quarter turn anticlockwise about (0, 0), which sends (1, 0) to (0, 1) and (0, 1) to (-1, 0)). The tests check that two flips and four quarter turns undo themselves.