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**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, a^{2} = b^{2} + c^{2}.

**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.

**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, a^{2} = b^{2} + c^{2}.

**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.