Question

given: x is the mid - point of (overline{wy}) and (overline{vz}) prove: (angle xwvcongangle xyz) statements: 1. x is the mid - point of (overline{wy}) and (overline{vz}) 2. (overline{wx}congoverline{yx}) 3. (overline{vx}congoverline{zx}) 4. (angle wxvcongangle yxz) 5. (\triangle xwvcong\triangle xyz) 6. (angle xwvcongangle xyz) reasons: 1. 2. 3. 4. 5. 6.

Explanation:

Step1: Given information

We are given that \(X\) is the mid - point of \(\overline{WY}\) and \(\overline{VZ}\).

Step2: Mid - point property

Since \(X\) is the mid - point of \(\overline{WY}\), by the definition of a mid - point, \(\overline{WX}\cong\overline{YX}\).

Step3: Mid - point property

Since \(X\) is the mid - point of \(\overline{VZ}\), by the definition of a mid - point, \(\overline{VX}\cong\overline{ZX}\).

Step4: Vertical angles

\(\angle WXV\) and \(\angle YXZ\) are vertical angles. Vertical angles are always congruent.

Step5: Triangle congruence

We have two pairs of congruent sides (\(\overline{WX}\cong\overline{YX}\) and \(\overline{VX}\cong\overline{ZX}\)) and a pair of congruent included angles (\(\angle WXV\cong\angle YXZ\)). So, by the Side - Angle - Side (SAS) congruence postulate, \(\triangle XWV\cong\triangle XYZ\).

Step6: CPCTC

Since \(\triangle XWV\cong\triangle XYZ\), then by the Corresponding parts of congruent triangles are congruent (CPCTC), \(\angle XWV\cong\angle XYZ\).

Answer:

1. Given
2. Definition of mid - point
3. Definition of mid - point
4. Vertical angles are congruent
5. Side - Angle - Side (SAS) congruence postulate
6. Corresponding parts of congruent triangles are congruent (CPCTC)