Question

In triangle ABC points M and N lie on sides AB and BC, respectively such that MN∥ AC . Segments AN and CM intersect at point K and AK = KC. Prove that △ABC is isosceles.

Answer 1

Statements Reasons

1.
`MN \parallel AC` 1. Given

2.
`AK = KC` 2. Given

3.
`m \angle NAC = m\angle MCA` 3. Base angles theorem.

4.
`m \angle NAC = m\angle MNA` 4. Alternate interior angles.

5.
`m \angle MCA = m \angle NMC` 5. Alternate interior angles.

6.
`MK = NK` 6. Isosceles triangle theorem.

7.
`m \angle MKA = m \angle NKC` 7. Vertical angles.

8.
`\triangle MKA \cong \triangle NKC` 8. By SAS postulate.

9.
`m \angle MAK = m \angle NCK` 9. By CPCTC.

10. 10. Sum of adjacent angles.

`m\angle MAK + m \angle NAC = m \angle NCK + m \angle MCA\\m \angle A = m \angle C`

11.
`\triangle ABC` is isosceles 11. Base angles theorem.