Use the definition of the angle bisector and look for a common side between both triangles.
Statements
Reasons
1.
WY bisects ∠ XWZ and ∠ XYZ
1.
Given
2.
∠ XWY ≅ ∠ ZWY
2.
Definition of Angle Bisector
3.
WY ≅ WY
3.
Reflexive Property of Congruent Segments
4.
∠ WYX ≅ ∠ WYZ
4.
Definition of Angle Bisector
5.
△ WXY ≅ △ WZY
5.
Angle-Side-Angle (ASA) Congruence Postulate
Practice makes perfect
Let's mark the congruent angles generated by the angle bisector WY.
By the Reflexive Property of Congruent Segments we have that WY ≅ WY. Next, we make a list of the congruent parts between both triangles.
cc
∠ XWY ≅ ∠ ZWY & Angle
WY ≅ WY & Included Side
∠ WYX ≅ ∠ WYZ & Angle
Consequently, by the Angle-Side-Angle (ASA) Congruence Postulate we obtain that △ WXY ≅ △ WZY. We can summarize this proof in the following two-column table.
