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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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