Question

Given that XW bisects AXZ, what theorem or theorems could be used to justify that W is equidistant from the two outside rays?

Answer 1

HA congruence postulate

Explanation:

In order to show W is equidistant from A and Z, we must demonstrate that WA ≅ WZ. The HA congruence postulate can be used for that purpose.

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proof

WX bisects AXZ, so ∠WXA≅∠WXZ by the definition of an angle bisector. By the reflexive property of congruence, WX≅WX. The angles at A and Z are given as right angles.

So, we have triangles WAX and WZX that are right triangles with congruent hypotenuse and corresponding angle. These are the preconditions for the application of the HA congruence postulate. That postulate says the triangles WAX and WZX are congruent, so ...

WA ≅ WZ by CPCTC. Hence W is equidistant from the outside rays.

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Additional comment

The definition of distance from a point to a line is the perpendicular distance from the point to the line. In this geometry, the perpendicular distances from W to the outside rays are the lengths of segments WA and WZ. That is why we need to show those lengths are the same.