Question

1. the congruency of \\( \triangle mno \\) and \\( \triangle xyz \\) can be proven using a reflection across the line bisecting \\( \overline{oz} \\). however, this congruency can also be proven using geometric postulates, theorems, and definitions. prove that the triangles are congruent using a two - column proof and triangle congruency theorems.

given: \\( \angle m \cong \angle x \\)
\\( \angle n \cong \angle y \\)
\\( \overline{yo} \cong \overline{nz} \\)

prove: \\( \triangle mno \cong \triangle xyz \\)

Explanation:

Step1: Use the Angle - Angle - Side (AAS) Congruence Theorem

The AAS theorem states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.
We are given that \(\angle M\cong\angle X\), \(\angle N\cong\angle Y\), and \(YO\cong NZ\).
Since \(YO + OZ=YZ\) and \(NZ + OZ = NO\) (by the segment addition postulate), and \(YO\cong NZ\) (given), we can conclude that \(NO\cong YZ\) (by substitution and the property of equality for segments).
Now, in \(\triangle MNO\) and \(\triangle XYZ\):

• \(\angle M\cong\angle X\) (given)
• \(\angle N\cong\angle Y\) (given)
• \(NO\cong YZ\) (proven above)

Answer:

By the Angle - Angle - Side (AAS) Congruence Theorem, \(\triangle MNO\cong\triangle XYZ\)