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# 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:**

We know that angles in the same segment are equal.

A diagram is constructed as per the given question.

It is given that BE is the bisector of ∠B, AD is the bisector of ∠A and CF is the bisector of ∠C.

Thus, ∠ABE = ∠B/2

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

Thus, ∠ADE = ∠B/2

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

∠D = ∠ADE + ∠ADF

= ∠B/2 + ∠C/2 [Since ∠ADE = ∠B/2 and ∠ADF = ∠C/2]

= 1/2 (∠B + ∠C )

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

= 90° - 1/2 A

Similarly, it can be proved for

∠E = 90° - 1/2 B

∠F = 90° - 1/2 C.

Thus we have proved that the angles of the triangle DEF are 90° - 1/2 A, 90° - 1/2 B, 90° - 1/2 C.

**☛ Check: **NCERT Solutions for Class 9 Maths Chapter 10

**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

**☛ Related Questions:**

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