Question

given: x is the mid - point of (overline{wy}), (overline{wx}=overline{xy}) prove: (overline{xy}congoverline{xz}) statements: 1. x is the mid - point of (overline{wy}) 2. (overline{wx}=overline{xy}) 3. (overline{wx}congoverline{xz}) 4. (overline{wx}=overline{xz}) 5. (overline{xy}=overline{xz}) 6. (overline{xy}congoverline{xz}) reasons: given given definition of midpoint definition of congruence substitution property definition of congruence

Explanation:

Step1: Recall mid - point definition

Since \(X\) is the mid - point of \(\overline{WY}\), by the definition of mid - point, \(WX = XY\) (Statement 2, Reason: Definition of Midpoint).

Step2: Given equality

We are given that \(WX=XZ\) (Statement 4, Given).

Step3: Substitute

Substitute \(WX\) with \(XY\) (from Statement 2) into \(WX = XZ\) (Statement 4). Since \(WX = XY\) and \(WX=XZ\), by the substitution property, \(XY = XZ\) (Statement 5).

Step4: Convert to congruence

By the definition of congruence (if two line - segments have the same length, they are congruent), if \(XY = XZ\), then \(\overline{XY}\cong\overline{XZ}\) (Statement 6, Reason: Definition of Congruence).

Answer:

The proof is completed as shown above with statements and corresponding reasons filled in the two - column proof table. The key steps are using the mid - point definition, given equalities, substitution, and the definition of congruence to prove \(\overline{XY}\cong\overline{XZ}\).