Question

y is the midpoint of \\( \overline { v x } \\) and \\( \overline { u w } \\) and \\( \overline { u x } \cong \overline { v w } \\). complete the proof that \\( \triangle u x y \cong \triangle w v y \\).

Explanation:

Step1: Given

Statement 1, 2, 3 are given in the problem.

Step2: Definition of mid - point

Since \(Y\) is the mid - point of \(\overline{VX}\), by the definition of mid - point (\(MY=\frac{1}{2}VX\) if \(M\) is the mid - point of \(VX\)), we have \(\overline{VY}\cong\overline{XY}\).

Step3: Definition of mid - point

Since \(Y\) is the mid - point of \(\overline{UW}\), by the definition of mid - point, we have \(\overline{UY}\cong\overline{WY}\).

Step4: SSS (Side - Side - Side) Congruence Criterion

In \(\triangle UXY\) and \(\triangle W VY\), we have \(\overline{UX}\cong\overline{VW}\) (given), \(\overline{VY}\cong\overline{XY}\) (from step 2), \(\overline{UY}\cong\overline{WY}\) (from step 3). So, by the SSS (Side - Side - Side) congruence criterion for triangles (\(SSS: If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent\)), \(\triangle UXY\cong\triangle W VY\).

Answer:

1. Given
2. Given
3. Given
4. Definition of mid - point
5. Definition of mid - point
6. SSS (Side - Side - Side) Congruence Criterion