Name – Ashwani Kumar

I. Proving the propertyIn figure 1, ABC is any triangle. Line (x y) is the line

parallel to (AC), passing through B.

    

B1 and A are alternate internals. The angle is the same, since lines

(x y) and (AC) are parallel. Therefore, B1 = A.

Likewise, B3 and C are the same. Therefore, B3 =

C. We know that B1 + B2 + B3 = 180°, since xBy is a straight

angle. From this we see that A + B + C = 180° in triangle ABC

In Euclidean geometry, the sum of the three angles of a triangle is 180°. How can we use this property when calculating the angles of a triangle?

II. Calculating the angles

A. In any triangle

Example: We want to calculate A of triangle ABC.

                             

                 

We apply the rule A + 114° + 25° = 180°.From this, we have the calculations: A + 139° = 180° and A = 180° - 139° = 41°.

B. In a right-angled triangle

The sum of the two acute angles in a right-angled triangle is 90°.Triangle ABC has a right angle at A.So: A = 90°. Therefore, 90° + B + C = 180°, which means B +

C = 90°.Example: We want to calculate B of triangle ABC, shown in figure 3, which has a right angle at A.

                                          

We apply the rule stated previously: B + C = 90°. From this, B + 57° = 90° and B = 90° - 57° = 33°. B is 33°.

C. In an isosceles triangleExample: We want to calculate A and B of

isosceles triangle ABC.

                                 

We apply the rule A + B + 48° = 180°. As ABC is an isosceles triangle in C, we know that A = B; therefore A + A + 48° = 180°, 2A + 48° = 180°,  2A = 180°– 48° = 132°, and A (and B ) is 66°.

D .In an equilateral triangleThe three angles of an equilateral triangle are

each 60°.

The angles are the same because the triangle is equilateral and their sum is 180°. Therefore, they are each degrees, i.e., 60°.