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
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}
Solution
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.