Question

#prove that the sum of interior angle of a triangle is 180 degree----

Answer 1

Theorem 1:Angle sum property of triangle states that, sum of interior angles of a triangle is 180°.

Proof:Consider a ∆ABC, as shown in the figure below.To prove the above property of triangles, draw a line PQ←→ parallel to the side BC of the given triangle.



Figure 2 Proof of angle sum property

Since PQ is a straight line, it can be concluded that:

∠PAB + ∠BAC + ∠QAC = 180°  ………(1)

SincePQ||BCand AB, AC are the transversals,

Therefore, ∠QAC = ∠ACB (pair of alternate angles)

Also, ∠PAB = ∠CBA(pair of alternate angles)

Substituting the value of ∠QAC and∠PAB in equation (1),

∠ACB + ∠BAC + ∠CBA= 180°

Thus, the sum of interior angles of a triangle is 180°

Answer 2

Statement : The sum of the angles of a triangle is 180°.

To prove : sum of the angles of ΔABC is 180°

Construction : Draw a line $lm$ parallel to BC.

Proof :

Since $lm$ || BC, we have ;

∠2 = ∠y ..... (i) . [Alternate interior angles]

Similarly,

∠1 = ∠z ....... (ii) . [Alternate interior angles]

Also, sum of angles at a point a on line $lm$ is 180°.

∴ ∠2 + ∠x + ∠1 = 180°

⇒ ∠y + ∠x + ∠z = 180°

⇒ ∠x + ∠y + ∠z = 180°

⇒ ∠A + ∠B + ∠C = 180°

Therefore,

Sum of all angles of a Δ is 180°