Question

18.
M is the midpoint of WX and
YZ. Is YW = ZX? Why?
W
M

Answer 1

Answer: Yes, YW ≅ ZX

Reasoning:

Because point M is the midpoint of segments WX and YZ, segments YM≅ZM and WM≅XM by the definition of a midpoint. Furthermore, ∠YMW≅∠ZMX by the Vertical Angles Theorem. Because we have a pair of corresponding, congruent sides, a pair of the included corresponding,congruent angles, and another pair of corresponding, congruent sides, the Side-Angle-Side Congruence Postulate can be used. By SAS, △YMW≅△ZMX. By CPCTC, segments YW≅ZX.

Answer 2

Yes, YW=ZX because triangles YMW and ZMX are congruent by SAS congruence postulate.

To prove that YW = ZX, we can use the SAS (Side-Angle-Side) Congruence Postulate. This postulate states that if two triangles have two congruent sides and the included angle is congruent, then the triangles are congruent.

In this case, we have two congruent sides, WM = MX and YM = MZ, and the included angle is congruent, Angle YMW = Angle ZMX (vertical angles). Therefore, triangles YMW and ZMX are congruent.

By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), we can conclude that YW = ZX.

Another way to think about this problem is to consider the following:

If we cut triangles YMW and ZMX out of the diagram and slide them so that their congruent sides overlap, the two triangles will completely coincide.

This means that the two triangles are identical in every way, including the length of their corresponding sides.

Therefore, YW must be equal to ZX.