Angle sum property of triangle — prove that angles add up to 180°

Question

Prove that the sum of all angles of a triangle is $180°$ . If two angles of a triangle are $65°$ and $45°$ , find the third angle.

(NCERT Class 7 — fundamental geometry theorem)

Solution — Step by Step

Let triangle $ABC$ have angles $\angle A$ , $\angle B$ , and $\angle C$ .

Draw a line $PQ$ through vertex $A$ , parallel to side $BC$ .

Now we have:

$PQ \parallel BC$ and $AB$ is a transversal
$PQ \parallel BC$ and $AC$ is a transversal

Since $PQ \parallel BC$ :

$\angle PAB = \angle ABC = \angle B$ (alternate interior angles, with $AB$ as transversal)
$\angle QAC = \angle ACB = \angle C$ (alternate interior angles, with $AC$ as transversal)

Points $P$ , $A$ , $Q$ lie on a straight line, so:

\angle PAB + \angle BAC + \angle QAC = 180°

Substituting:

\angle B + \angle A + \angle C = 180°

Therefore, the sum of angles of a triangle is $180°$ .

Given: $\angle 1 = 65°$ , $\angle 2 = 45°$

\angle 3 = 180° - 65° - 45° = \mathbf{70°}

Why This Works

The proof hinges on two facts: (1) alternate interior angles are equal when a transversal crosses parallel lines, and (2) angles on a straight line add up to $180°$ . By drawing a parallel line through one vertex, we “transfer” the two base angles to sit next to the top angle — and together they form a straight line.

This is one of the most elegant proofs in geometry because it connects the triangle property to the simpler property of parallel lines.

Alternative Method — Exterior Angle Approach

Another way to see this: the exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

If we extend side $BC$ beyond $C$ , the exterior angle at $C$ = $\angle A + \angle B$ .

Since the interior angle at $C$ + exterior angle at $C$ = $180°$ (linear pair):

\angle C + (\angle A + \angle B) = 180°

\angle A + \angle B + \angle C = 180°

For CBSE exams: the proof using the parallel line is the standard NCERT proof. Write it step-by-step with a clear diagram. Always mention “alternate interior angles” and “angles on a straight line” — these are the reasons the examiner looks for. Without stating the reasons, you’ll lose marks even if the logic is correct.

Common Mistake

Students often skip the construction step (drawing the parallel line) and try to prove it without any construction. The proof doesn’t work without the parallel line — you need it to create the alternate interior angles. Also, when finding the third angle, some students subtract only one angle from $180°$ and forget to subtract the second. Always write: third angle = $180° - \text{(first angle)} - \text{(second angle)}$ .

Question

Solution — Step by Step

Why This Works

Alternative Method — Exterior Angle Approach

Common Mistake

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