Notes On Properties of Triangles - CBSE Class 7 Maths
Angle sum property The sum of the three angles of a triangle is 180Â°. e.g. If A, B and C are the angles of a triangle ABC, then âˆ A + âˆ B + âˆ C = 180Â°. Proof: Consider a triangle ABC. Let line XY be parallel to side BC at A. AB is a transversal that cuts the line XY and AB, at A and B, respectively.  As the alternate interior angles are equal, âˆ 1 = âˆ 4 and âˆ 2 = âˆ 5.   âˆ 4, âˆ 3 and âˆ 5 form linear angles, and their sum is equal to 180Â°. â‡’ âˆ 4 + âˆ 3 + âˆ 5 = 180Â° â‡’ âˆ 1 + âˆ 2 + âˆ 3 = 180Â° Hence, the sum of the three angles of a triangle is 180Â°. Exterior angle property An exterior angle of a triangle is equal to the sum of its opposite interior angles. e.g. If âˆ 4 is an exterior angle of Î”ABC,  âˆ 1 and âˆ 2 are the interior opposite angles, then âˆ 4 = âˆ 1 + âˆ 2. The sum of the lengths of any two sides of a triangle is greater than the third side. In a right-angled triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called its legs. Pythagorean theorem In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.   If b and c are legs and a is the hypotenuse of a right angled triangle then, a2 = b2 + c2. Converse of Pythagorean theorem If the sum of the squares on two sides of a triangle is equal to the square of the third side, then the triangle must be a right-angled triangle.

#### Summary

Angle sum property The sum of the three angles of a triangle is 180Â°. e.g. If A, B and C are the angles of a triangle ABC, then âˆ A + âˆ B + âˆ C = 180Â°. Proof: Consider a triangle ABC. Let line XY be parallel to side BC at A. AB is a transversal that cuts the line XY and AB, at A and B, respectively.  As the alternate interior angles are equal, âˆ 1 = âˆ 4 and âˆ 2 = âˆ 5.   âˆ 4, âˆ 3 and âˆ 5 form linear angles, and their sum is equal to 180Â°. â‡’ âˆ 4 + âˆ 3 + âˆ 5 = 180Â° â‡’ âˆ 1 + âˆ 2 + âˆ 3 = 180Â° Hence, the sum of the three angles of a triangle is 180Â°. Exterior angle property An exterior angle of a triangle is equal to the sum of its opposite interior angles. e.g. If âˆ 4 is an exterior angle of Î”ABC,  âˆ 1 and âˆ 2 are the interior opposite angles, then âˆ 4 = âˆ 1 + âˆ 2. The sum of the lengths of any two sides of a triangle is greater than the third side. In a right-angled triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called its legs. Pythagorean theorem In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.   If b and c are legs and a is the hypotenuse of a right angled triangle then, a2 = b2 + c2. Converse of Pythagorean theorem If the sum of the squares on two sides of a triangle is equal to the square of the third side, then the triangle must be a right-angled triangle.

Next
âž¤