P is a point on the bisector of ∠ ABC. If the line through P, parallel to BA meet BC at Q, prove that BPQ is an isosceles triangle.
consider the figure above.
P is the point on the bisector of angle ABC.
∠1 =∠ 2 [ BP is the bisector of ∠ABC]
Also PQ is parallel to AB
∠1 = ∠3 [ If a transversal intersects two parallel lines, then alternate interior angles are equal ]
∴∠ 2 = ∠3
And since in a triangle if two angles are equal then its sides opposite to these angles are also equal.
∴BQ = PQ
Now, two sides of the ?BPQ are equal.
∴ΔBPQ is isosceles.