# Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle DEF are 90° - 1/2 A, 90° - 1/2 B, 90° - 1/2 C

**Solution:**

Angles in the same segment are equal.

It is given that BE is the bisector of ∠B.

∠ABE = ∠B/2

However, ∠ADE = ∠ABE (Angles in the same segment for chord AE)

∠ADE = ∠B/2

Similarly, ∠ADF = ∠ACF = ∠C/2 (Angle in the same segment for chord AF)

∠D = ∠ADE + ∠ADF

= ∠B/2 + ∠C/2

= 1/2 (∠B + ∠C )

= 1/2 (180° - ∠A) [Angle sum property of a triangle]

= 90° - 1/2 A

Similarly, it can be proved for

∠E = 90° - 1/2 B

∠F = 90° - 1/2 C.

**Video Solution:**

## Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle DEF are 90° - 1/2 A, 90° - 1/2 B, 90° - 1/2 C

### Maths NCERT Solutions Class 9 - Chapter 10 Exercise 10.6 Question 8:

**Summary:**

If the bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. We have proved that the angles of the triangle DEF are 90° - 1/2 A, 90° - 1/2 B, 90° - 1/2 C