In the diagram, which distances would need to be the same in order for point A to be on the bisector of angle W?

Question

 

 

AS=AZ 

 

AP=AX

 

AR=AY

All of the above.

Answer 1

 

This one is a bit tricky.   Notice that the points P, X, S, and Z  look to be symmetrical but there is nothing to indicate that, for example, the lengths PW and XW are equal; therefore, if the lengths AP and AX are equal it tells us nothing.   Lines AR and AY however come off at right angles, and if the lengths AR and AY are equal, the point A would have to lie on the bisector of angle W.

 

So AR=AY is the only valid choice.   

 

Answer 2

From the diagram AR ⊥ RW and AY ⊥ YW.

 

This means that ∠ARW  ≅ ∠AYW, since they are right angles.

 

Now when AR = AY, then these would be congruent sides of right triangles ΔARW and ΔAYW

 

Also AW is the hypotenuse of both ΔARW and ΔAYW.

 

∴ ΔARW ≅ ΔAYW, by the Hypotenuse-Leg theorem.

 

∠AWR ≅ ∠AWY since corresponding parts of congruent triangles are congruent.

 

This then means that AW bisects angle ∠W.