**Question:**

In △ABC, M and N are points on AB and AC respectively such that BM = CN. If ∠B = ∠C then show that MN∥BC.

**Solution:**

In △ABC, ∠B = ∠C

∴AB = AC (Sides opposite to equal angle are equal)

Subtracting BM from both sides, we get

AB − BM = AC − BM

⇒AB − BM = AC − CN (∵BM = CN)

⇒AM = AN

∴∠AMN = ∠ANM (Angles opposite to equal sides are equal)

Now, In △ABC,

∠A + ∠B + ∠C = 180o .....(1)

(Angle Sum Property of triangle)

Again In In △AMN,

∠A +∠AMN + ∠ANM = 180o ......(2)

(Angle Sum Property of triangle)

From (1) and (2), we get

∠B + ∠C = ∠AMN + ∠ANM

⇒2∠B = 2∠AMN

⇒∠B = ∠AMN

Since, ∠B and ∠AMN are corresponding angles.

∴ MN∥BC

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