Question

Prove that sum of the three angles of a triangle is two right angles or 180°
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Answer 1

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I have two proofs to prove it...

Proof 1

Let △ABC be a triangle.

Let BC be extended to a point D.

From External Angle of Triangle equals Sum of other Internal Angles:

∠ACD=∠ABC+∠BAC∠ACD=∠ABC+∠BAC

Bby by Euclid's Second Common Notion:

∠ACB+∠ACD=∠ABC+∠BAC+∠ACB∠ACB+∠ACD=∠ABC+∠BAC+∠ACB

But from Two Angles on Straight Line make Two Right Angles, ∠ACB+∠ACD∠ACB+∠ACD equals two right angles.

So by Euclid's First Common Notion, ∠ABC+∠BAC+∠ACB∠ABC+∠BAC+∠ACB equals two right angles.

Proof 2

Let ΔABCΔABC be a triangle.

Let DAEDAE be a line such that DE∥BCDE∥BC.

By Parallelism implies Equal Alternate Angles, ∠DAB=∠ABC∠DAB=∠ABC and ∠EAC=∠ACB∠EAC=∠ACB.

Therefore, the sum of the three angles is ∠ABC+∠BCA+∠CAB=∠DAB+∠BAC+∠CAE=180∘∠ABC+∠BCA+∠CAB=∠DAB+∠BAC+∠CAE=180∘.

❥Hope it helps!!!