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Transformation Geometry Proofs : Reflections

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Let l, m, n be distinct lines and P, Q, R be distinct points. Prove the following:
(a) sigma l sigma m = sigma m sigma l if and only if l perpendicular to m.
(b) sigma p sigma m = sigma m sigma p if and only if P E m.
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Proofs involving reflections are provided. The solution is detailed and well presented.

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a. If , then we consider a point . We have

So we get . But we know by definition of and . Therefore, we have .
If , we can suppose is the x-axis and is the y-axis. For any point , we have , . Therefore, we get .
b. We can suppose is the x-axis and . We know for any , we have
If , then for any point , we ...

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