a From the information in the exercise, we know that plane P is a perpendicular bisector of XZ at point Y. This means XY≅ YZ and that WY is perpendicular to XZ as it runs along plane P. Let's add this information to the exercise.
According to the Perpendicular Bisector Theorem, in a plane, if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Using this theorem, we can prove that W is equidistant from X and Z.
b Just like WY was a part of plane P, so is YV. Therefore, by to the Perpendicular Bisector Theorem, we can also prove that XV≅ ZV.
c Let's mark the two angles in our diagram.
We see that △ VXW contains ∠ VXW and △ VZW contains ∠ VZW. From part A and part B, we also know that
VX≅ VZ and XW≅ ZW
Since △ VXW and △ VZW share VW as a side, we can by the Reflexive Property of Congruence claim that these sides are congruent. Using the SSS Congruence Theorem we can show that our triangles are congruent. Since ∠ VXW and ∠ VZW are corresponding angles, we can prove that
∠ VXW ≅ ∠ VZW.
