Questions: W r bisects ∠X W Z. Complete the proof that W Z ≅ W x. Statement Reason ------ W Y bisects ∠X W Z Given ∠Z ≅ ∠X Given ∠X W Y ≅ ∠Y W Z Definition W Y ≅ W Y Reflexive △W X Y ≅ △W Z Y AAS W Z ≅ W X

W r bisects ∠X W Z. Complete the proof that W Z ≅ W x.

Statement  Reason
------
W Y bisects ∠X W Z  Given
∠Z ≅ ∠X  Given
∠X W Y ≅ ∠Y W Z  Definition
W Y ≅ W Y  Reflexive
△W X Y ≅ △W Z Y  AAS
W Z ≅ W X
Transcript text: $\overleftrightarrow{W r}$ bisects $\angle X W Z$. Complete the proof that $\overline{W Z} \cong \overline{W x}$. \begin{tabular}{|l|l|l|} \hline & Statement & Reason \\ \hline 1 & $\overleftrightarrow{W Y}$ bisects $\angle X W Z$ & Given \\ 2 & $\angle Z \cong \angle X$ & Given \\ 3 & $\angle X W Y \cong \angle Y W Z$ & Definition \\ 4 & $\overline{W Y} \cong \overline{W Y}$ & Reflexive \\ 5 & $\triangle W X Y \cong $\triangle W Z Y$ & AAS \\ 6 & $\overline{W Z} \cong \overline{W X}$ & \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Identify the given information

We are given that WY bisects ∠XWZ and ∠Z ≅ ∠X.

Step 2: Use the definition of angle bisector

Since WY bisects ∠XWZ, it divides it into two congruent angles. Therefore, ∠XWY ≅ ∠YWZ.

Step 3: Use the reflexive property

The segment WY is common to both triangles ΔWXY and ΔWZY. Thus, WY ≅ WY by the reflexive property of congruence.

Step 4: Apply the AAS congruence theorem

We have two pairs of congruent angles (∠XWY ≅ ∠YWZ and ∠Z ≅ ∠X) and a pair of congruent non-included sides (WY ≅ WY). Therefore, ΔWXY ≅ ΔWZY by the Angle-Angle-Side (AAS) Congruence Theorem.

Step 5: Corresponding parts of congruent triangles are congruent (CPCTC)

Since ΔWXY ≅ ΔWZY, their corresponding parts are congruent. Therefore, WZ ≅ WX.

Final Answer:

WZ ≅ WX is proven by the AAS Congruence Theorem and CPCTC.

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