Final answer:
The question requires a proof involving congruent triangles and the application of the Side-Angle-Side postulate combined with CPCTC to show that YZ equals WZ, given that ZX bisects angle YXW and YX equals WX.
Step-by-step explanation:
The question presents a geometrical proof situation where the given information is that we have a line segment ZX that bisects angle YXW, and that YX = WX. From these details, we are to prove that YZ equals WZ.
Using the properties of geometric figures and trigonometry, specifically those that deal with congruent triangles and bisected angles, we can determine the steps necessary to reach the desired conclusion. Notably, bisected angles indicate that the two angles adjacent to the bisector are congruent. Combined with the given that YX = WX, we would use the Side-Angle-Side (SAS) postulate to prove that the triangles formed are congruent. Once the congruence is established, we can conclude that corresponding parts of congruent triangles are equal, hence YZ = WZ.
The proof would involve identifying the two triangles that share angle YXW and ZX as their common side, and using the congruent segments and angles to prove that the triangles themselves are congruent. Once that is done, it follows by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem that YZ = WZ, since they are corresponding sides of the proved congruent triangles.