QuestionQuestion

Point P is given to be in the interior of angleXYZ. Prove:
(a) If P is the same distance away from the sides of angleXYZ, then P must lie on the angle bisector of angleXYZ.
(b) If P lies n the angle bisector of angleXYZ, then P is the same distance away from the sides of angleXYZ.

Given hint: the explanation part of these proofs is not hard; you're looking for one simple triangle congruence to get the job done in each part. However, the setup of each proof requires some care. In order to understand what is given, you need to recall what it means for a point to be a certain distance away from a line.

Each statement is the converse of the other, so we could write these two statements as one nice if and only if statement if we felt like it.
Proofs With Angles

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    Proof: a) We need to show that the segment PY makes equal angles to the segments XY and ZY.
    Drop the perpendicular from P to the segment XY. We obtain a point E on XY and PE is perpendicular to XY. Drop the perpendicular from P to the segment XZ....
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