hey if line bisect ext. angle C then how is it parellel to AC .can u write the question again...

Given Information
- Triangle ABC with exterior angle at vertex C.
- The bisector of the exterior angle C is parallel to line AC.
To Prove
- Triangle ABC is isosceles (AB = BC).
Proof Steps
1. Understanding the Geometry
- Let D be the point where the bisector of the exterior angle at C intersects the extension of line AC.
- Since the bisector of the exterior angle is parallel to AC, we have the following relationships based on Alternate Interior Angles.
2. Using Angle Relationships
- Since CD (the bisector) is parallel to AC, we know that:
- Angle ACD = Angle DCB (alternate interior angles).
- Let angle ACB = x. Then, angle ACD = 180 - x (exterior angle).
3. Applying the Angle Bisector Theorem
- The angle bisector divides the exterior angle into two equal parts, hence:
- Angle DCB = Angle ACB = x.
- Therefore, angle DCB = 180 - x.
4. Establishing Equal Angles
- Since ACD = DCB, we have:
- 180 - x = x.
- Solving this gives us:
- 2x = 180, or x = 90 degrees.
5. Conclusion about Triangle ABC
- From the above, we find that angles ACB and DBC are equal.
- Thus, AB = BC (since angles opposite to equal sides in a triangle are equal).
- Therefore, triangle ABC is isosceles.
Final Statement
- Hence, if the bisector of an exterior angle at C is parallel to AC, triangle ABC is indeed isosceles, proving AB = BC.