Question

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

Answer 1

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 =180⁰ ----(1) 

 (Angle Sum Property of triangle) 

Again In ∆AMN, 

∠A + ∠AMN + ∠ ANM =180⁰ ----(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