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Angle Sum Property of a Triangle

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Theorem: The sum of the angles of a triangle is 180º.
Proof :  Let us see what is given in the statement above, that is, the hypothesis and what we need to prove. We are given a triangle PQR and ∠ 1, ∠ 2 and ∠ 3 are the angles of ∆ PQR in following fig . 

We need to prove that ∠ 1 + ∠ 2 + ∠ 3 = 180°. Let us draw a line XPY parallel to QR through the opposite vertex P, as shown in Fig. 

Now, XPY is a line.
Therefore, ∠ 4 + ∠ 1 + ∠ 5 = 180° (1)
But XPY || QR and PQ, PR are transversals.
So, ∠ 4 = ∠ 2 and ∠ 5 = ∠ 3 (Pairs of alternate angles)
Substituting ∠ 4 and ∠ 5 in (1), we get
∠ 2 + ∠ 1 + ∠ 3 = 180°
That is, ∠ 1 + ∠ 2 + ∠ 3 = 180°

Theorem : If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
It is obvious from the above theorem that an exterior angle of a triangle is greater than either of its interior apposite angles.

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Theorem: The sum of the angles of a triangle is 180º. [00:07:13]
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