How to Prove the Angle Sum Property of a Triangle

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 :

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