Question
Prove that the sum of all interior angles of a triangle is 180°.
This is a fundamental theorem in Euclidean geometry — NCERT Class 9, Chapter 6. Every board exam student must know this proof cold.
Solution — Step by Step
Take triangle ABC. Through vertex A, draw a line PQ parallel to the base BC.
We need this parallel line because the entire proof depends on alternate interior angles — which only appear when a transversal cuts parallel lines.
Line PQ passes through A. On one side of A, we have point P; on the other, point Q.
Notice that angles ∠PAB, ∠BAC, and ∠CAQ sit together on line PQ at point A. Since PQ is a straight line:
AB is a transversal cutting the parallel lines PQ and BC.
By the Alternate Interior Angles theorem: (i.e., ).
Why alternate angles? Because ∠PAB and ∠ABC are on opposite sides of the transversal AB, between the parallel lines PQ and BC.
AC is another transversal cutting PQ and BC.
Similarly: (i.e., ).
Both pairs are alternate interior angles — same logic, different transversal.
Replace in the equation from Step 2:
Hence proved. The sum of all interior angles of a triangle is 180°.
Why This Works
The proof rests entirely on two pillars: the straight angle (180°) and alternate interior angles. We “borrowed” a straight line at vertex A to create a container that holds all three angles of the triangle.
The key insight is that when two parallel lines are cut by a transversal, alternate interior angles are equal. By drawing PQ ∥ BC, both sides AB and AC become transversals — and each one “transfers” one base angle up to point A.
This is why the theorem is true in Euclidean geometry specifically. On a curved surface (like the Earth), the angle sum of a triangle is greater than 180° — but that’s far beyond Class 9 scope.
Alternative Method
Using exterior angle property (for verification, not a standard proof):
We know the exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
For triangle ABC, extend BC to D. Then:
But (linear pair).
So .
This is more of a consistency check than an independent proof — it uses a result that itself needs the parallel line proof to establish.
In board exams, always state “by alternate interior angles (PQ ∥ BC, AB is transversal)” explicitly. Examiners deduct marks if you write the angles as equal without giving the reason.
Common Mistake
Saying “∠PAB = ∠ABC because PQ ∥ BC” without naming the transversal.
The reason must be: “Alternate interior angles, since PQ ∥ BC and AB is the transversal.” Naming the transversal is mandatory. Without it, you’ve stated a fact with an incomplete justification — CBSE boards will dock 1 mark for this every time.
Also, don’t confuse alternate interior angles with co-interior (same-side interior) angles. Alternate angles are equal; co-interior angles are supplementary (add to 180°). These are opposite results, and mixing them up collapses the entire proof.