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

Basic Proportionality theorem could be used to justify that equidistant from the two outside rays.

What is Basic Proportionality Theorem?

Thales's theorem, basic proportionality theorem or side splitter theorem is an important theorem in elementary geometry about the ratios of various line segments that are created if two intersecting lines are intercepted by a pair of parallels.

Here, Given that XW bisects AXZ

Thus, Basic Proportionality theorem could be used to justify that equidistant from the two outside rays.

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Answer 2

Answer:

  HA congruence postulate

Step-by-step 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.