Question

10. Given: YX = ZX, WX bisects /YXZ Prove: AWYX = AWZX

Answer 1

Given YX congruent to ZX and WX bisecting
`\( \angle YXZ \)`, triangles WYX and WZX are congruent by Side-Angle-Side (SAS) congruence, establishing equality in both sides and angles.

Statements and Reasons:

1. Given: YX is congruent to ZX.

- Reason: Given information.

2. Given: WX bisects angle YXZ.

- Reason: Given information.

3. Definition of Angle Bisector:

-
`\( \angle WYX \) and \( \angle WZX \)` are formed by WX, the angle bisector of
`\( \angle YXZ \).`

- Reason: By the definition of an angle bisector.

4. Side-Angle-Side (SAS) Congruence:

- YX is congruent to ZX (Given).

- WX is the common side.

-
`\( \angle WYX \)` is congruent to
`\( \angle WZX \)` (Angle bisector).

- Reason: Using SAS congruence.

5. Conclusion: Triangle Congruence:

-
`\( \triangle WYX \)` is congruent to
`\( \triangle WZX \).`

- Reason: From statement 4, triangles WYX and WZX are congruent by SAS congruence.

Therefore, the given conditions imply that
`\( \triangle WYX \)` is congruent to
`\( \triangle WZX \).`

Answer 2

Here, we want to proof that the two triangles are congruent

a) Statement 1

YX = ZX

Reason 1

Given

b) Statement 2

angle WXZ = angle WXY

Reason 2

Given that WX bisects angle YXZ, the two angles are equal

c) Statement 3

WX = WX

Reason 3

Reflexive property of equality

d) Statement 4

Triangle WYX is congruent to Triangle WZX

Reason 4

Side-angle-side SAS