Question

z is the midpoint of \overline{xy}. complete the proof that \angle xwz \cong \angle ywz.

1 z is the midpoint of \overline{xy}
2 \overline{wy} \cong \overline{wx}
3 \overline{xz} \cong \overline{yz}
4 \overline{wz} \cong \overline{wz}
5 \triangle wxz \cong \triangle wyz
6 \angle xwz \cong \angle ywz

Explanation:

Step1: Given

Given in the problem statement.

Step2: Given

Given in the diagram (marked with tick marks).

Step3: Definition of midpoint

If \(Z\) is the mid - point of \(\overline{XY}\), then \(XZ = YZ\) (by the definition of a mid - point, which divides a segment into two equal parts, so \(\overline{XZ}\cong\overline{YZ}\)).

Step4: Reflexive property of congruence

For any segment \(\overline{a}\), \(\overline{a}\cong\overline{a}\). Here \(a = WZ\), so \(\overline{WZ}\cong\overline{WZ}\).

Step5: SSS (Side - Side - Side) congruence criterion

If in two triangles \(\triangle WXZ\) and \(\triangle WYZ\), \(\overline{WX}\cong\overline{WY}\), \(\overline{XZ}\cong\overline{YZ}\), and \(\overline{WZ}\cong\overline{WZ}\), then \(\triangle WXZ\cong\triangle WYZ\) (by SSS, which states that if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent).

Step6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle WXZ\cong\triangle WYZ\), their corresponding angles \(\angle XWZ\) and \(\angle YWZ\) are congruent.

Answer:

1. Given
2. Given
3. Definition of mid - point
4. Reflexive property of congruence
5. SSS (Side - Side - Side)
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)