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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.

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olayemiolakunle65

Answer:

Step-by-step explanation:

An isosceles triangle has two equal sides and two equal base angles.

Given that : AK = KC and MN∥ AC

ΔMNB is similar to ΔACB (congruence property)

ΔANB = ΔANC (SAS congruence property)

ΔCMB = ΔCMA (SAS congruence property)

[tex]\frac{AB}{BN}[/tex] = [tex]\frac{AC}{CN}[/tex] (Angle bisector theorem)

So that:

<BAC = <ACB (base angle property of an isosceles triangle)

/AB/ = /BC/ (side property of an isosceles triangle)

Therefore, ΔABC is an isosceles triangle.

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