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**Angle Sum Property of a Triangle:**

The sum of the measures of three angles of a triangle is 180°.

**Proof of Angle Sum Property of a Triangle**

Given : A triangle ABC.

To prove : *∠ A +∠B +∠C = *180*°*

Construction : Draw a line segment PQ through A and parallel to BC.

** Proof: ** Mark the angles as indicated in the figure.

Hence, *∠ A +∠B +∠C = *180*°*

or sum of the angles of a triangle is 180°

**ANGLE BISECTORS OF A TRIANGLE AND IN-CENTRE
Angle Bisectors :-**

Angle bisector of a triangle is a line segment which bisect the angle and whose end points lies on the vertex and its opposite side.

• In the given figure, AD is the bisector of

So *∠*BAD* = ∠*DAC

** Incentre **

• In-centre is the meeting point of all three angle bisectors of a triangle.

• In the given figure, I is the Incentre of *∆*

• In-centre always lies inside the triangle.

• The perpendicular distance from incentre to the side of triangle is always same.

*⇒* IP = IQ = IR

• A circle can be drawn inside the triangle by assuming I as centre and IP as radius.

**PERPENDICULAR BISECTOR AND CIRCUMCENTRE
Perpendicular Bisector **

It is a line segment which is perpendicular to the side of a triangle and passing through the mid point of the side.

In the given figure MN is the perpendicular bisector for *∆*

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