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दम्भो दर्पोऽभिमानश्च क्रोध: पारुष्यमेव च |अज्ञानं चाभिजातस्य पार्थ सम्पदमासुरीम् ||

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.

Angle-Sum-Property-of-a-Triangle-theorem-1

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.

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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.

Angle-Sum-Property-of-a-Triangle-theorem-2

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.
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