**Angle Sum Property of a Triangle**

**Theorem 1:**

Prove that sum of all three angles is 180° or 2 right angles.

**Given:** ∆ABC

**To prove:** ∠A + ∠B + ∠C = 180°

**Construction:** Draw PQ || BC, passes through point A.

**Proof:** ∠1 = ∠B and ∠3 = ∠C ……. (i)

[∵ alternate angles ∵ PQ || BC]

∵ PAQ is a line

∴∠1 + ∠2 + ∠3 = 180° (linear pair application)

∠B + ∠2 + ∠C = 180°

∠B + ∠CAB + ∠C = 180°

= 2 right angles.

Proved.

**Read More:**

- Median and Altitude of a Triangle
- The Angle of An Isosceles Triangle
- Areas of Two Similar Triangles
- Area of A Triangle
- To Prove Triangles Are Congruent
- Criteria For Similarity of Triangles
- Construction of an Equilateral Triangle
- Classification of Triangles

**Theorem 2:**

If one side of a triangle is produced then the exterior angle so formed is equal to the sum of two interior opposite angles.

Means ∠4 = ∠1 + ∠2

Proof : ∠3 = 180° – (∠1 + ∠2) ….(1)

(by angle sum property)

and BCD is a line

∴ ∠3 + ∠4 = 180° (linear pair)

or ∠3 = 180° – ∠4 …..(2)

by (1) & (2)

180° – (∠1 + ∠2) = 180° – ∠4

⇒ ∠1 + ∠2 = ∠4 Proved.

**Note :**

- Each angle of an equilateral triangle measures 60º.
- The angles opposite to equal sides of an isosceles triangle are equal.
- A scalene triangle has all angles unequal.
- A triangle cannot have more than one right angle.
- A triangle cannot have more than one obtuse angle.
- In a right triangle, the sum of two acute angles is 90º.
- The sum of the lengths of the sides of a triangle is called perimeter of triangle.