The triangle interior angle sum theorem states that the sum of all interior angles in any triangle equals 180 degrees. This fundamental property applies to all triangles regardless of their shape or size - whether they are acute, right, or obtuse triangles.
To understand the triangle angle sum proof, we need to establish key properties of parallel lines and transversals. When a transversal line crosses two parallel lines, it creates several important angle relationships. Alternate interior angles are equal, and corresponding angles are equal. These properties will be essential for our proof.
Now we construct the key element for our proof. Starting with triangle ABC, we draw a line through vertex A that is parallel to side BC. This parallel line construction is crucial because it creates the geometric relationships we need to prove that the triangle's interior angles sum to 180 degrees.
Now we identify the crucial angle relationships created by our parallel line construction. The parallel line through vertex A creates alternate interior angles with the triangle's sides. Angle ABC equals angle BAL, and angle ACB equals angle CAR. These equal angle pairs are the key to our proof.
Now we complete the proof. The three angles at vertex A on the straight line sum to 180 degrees. Since we proved that angle BAL equals angle ABC and angle CAR equals angle ACB, we can substitute these equal angles. This gives us angle ABC plus angle BAC plus angle ACB equals 180 degrees, proving that the sum of interior angles in any triangle is 180 degrees.