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Flag and two parallel lines

You might like to copy the diagram above onto squared paper to help you with this problem.

Reflect the flag in the left-hand line, and then reflect the image you obtain in the right-hand line. Can you describe the single transformation that takes the first flag to the last flag?

Repeat this with flags in different positions and orientations, and with different pairs of parallel lines.

Does it matter in which line you reflect first?
Does it matter whether the flag is to one side of the lines or between the lines?

Try to find a way to describe the single transformation for a general pair of parallel lines.
Can you prove your answer?

If you have enjoyed this problem, why not have a go at ...on the Wall and Who is the fairest of them all? ?