Question

Prove that the sum of the angles of a triangle is 180
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Answer 1

Figure :

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From figure

Given :

• ABC is a traingle.

To Prove :

• ∠1 + ∠2 + ∠3 = 180°.

Construction :

• Draw LM || BC which passes through A.

Proof :

∵ LM || BC and AB is a transversal then ∠5 = ∠2 ....(❶)

And, AC is transversal then ∠4 = ∠3 ....(❷)

Now,

⟶ ∠5 + ∠1 + ∠4 = 180° [ linear pair ]

From equation (❶) and equation (❷), we get

⟶ ∠1 + ∠2 + ∠3 = 180°.

Hence, the sum of the angles of a triangle is 180°.

$\:$

Alternative method :

Figure :

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From figure

Given :

• ABC is a traingle.

To Prove :

• ∠ABC + ∠ACB + ∠BAC = 180°.

Construction :

• Draw a line passing through A parallel to BC.

Proof :

⟶ ∠ABC = ∠XAB [ Alternate interior angle ] ....(❶)

⟶ ∠ACB = ∠YAC [ Alternate interior angle ] ....(❷)

⟶ ∠XAB + ∠BAC + ∠YBC [ linear pair ]

From equation (❶) and equation (❷), we get

⟶ ∠ABC + ∠ACB + ∠BAC = 180°.

Hence, the sum of the angles of a triangle is 180°.

Hence Proved !

Answer 2

Answer:

As the Question is asked to proof that the sum of the angles of a triangle is 180°.

One of the first things we all learned about triangles is that the sum of the interior angles is 180 degrees.

How do we know that the sum of the angles is always 180? Is there some way that we can definitively prove it?

• The answer is yes!

To mathematically prove that the angles of a triangle will always add up to 180 degrees, we need to Construct it . Let us do the solution.

Given:

• △ABC → is a triangle with angles ∠ a, ∠ b, ∠ c

To Prove:

• △ABC → ∠ a + ∠ b + ∠ c = 180°

We know that the sum of angles within a triangle is 180° .

Here to prove this thing, we need to do a small bit of construction.

Construction:

First we have to draw a line 'PQ', passing through the point 'C' and which is parallel to 'AB'.

• Now ∠ a = ∠ 1 (i) (alternate angles AB || PQ )

• And ∠ b = ∠ 2 (ii) (alternate angles AB || PQ )

• Now ∠ 1 + ∠ 2 + ∠ c = 180° (leaner angles)

Now by the equation number 1, 2 and 3

∠ a + ∠ b + ∠ c = 180° Hence proved.