Question

given: \\( \overline { wy } \\) bisects \\( \angle xwz \\) \\( \overline { wy } \perp \overline { xz } \\) prove: (a) \\( \delta ywz \cong \delta ywx \\) (b) \\( \overline { xy } \cong \overline { zy } \\)

Explanation:

Step1: Identify the given information

We know that \(WY\) bisects \(\angle XWZ\), so \(\angle XWY=\angle ZWY\). Also, \(WY\perp XZ\), so \(\angle WYX = \angle WYZ=90^{\circ}\). And \(WY = WY\) (common side).

Step2: Prove \(\triangle YWZ\cong\triangle YWX\)

By the Angle - Angle - Side (AAS) congruence criterion. In \(\triangle YWZ\) and \(\triangle YWX\):

• \(\angle XWY=\angle ZWY\) (angle)
• \(\angle WYX=\angle WYZ\) (angle)
• \(WY = WY\) (side)

So, \(\triangle YWZ\cong\triangle YWX\) (AAS).

Step3: Prove \( \overline{XY}\cong\overline{ZY}\)

Since \(\triangle YWZ\cong\triangle YWX\), corresponding parts of congruent triangles are congruent. So, \(XY = ZY\) (by CPCTC - Corresponding Parts of Congruent Triangles are Congruent), which means \(\overline{XY}\cong\overline{ZY}\).

Answer:

(a) \(\triangle YWZ\cong\triangle YWX\) by AAS. (b) \(\overline{XY}\cong\overline{ZY}\) by CPCTC.