Proof that the sum of triangle is 180 0

Question

Proof that the sum of triangle is 180 0

Asked on March 29, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-29T16:09:55.091Z

Let's prove that the sum of angles in a triangle is 180 degrees. Step 1: Consider a triangle ABC. Let the angles of the triangle be ∠A, ∠B, and ∠C. Step 2: Draw a line parallel to one side. Draw a line XY passing through vertex A such that XY is parallel to the side BC. X-------A-------Y / \ / \ / \ / \ B-----C Step 3: Identify alternate interior angles. Since XY is parallel to BC, and AB is a transversal line: The angle ∠XAB is an alternate interior angle to ∠ABC (which is ∠B). Therefore, ∠XAB = ∠B. Similarly, since XY is parallel to BC, and AC is a transversal line: The angle ∠YAC is an alternate interior angle to ∠ACB (which is ∠C). Therefore, ∠YAC = ∠C. Step 4: Sum the angles on the straight line. The angles on the straight line XY at point A are ∠XAB, ∠BAC (which is ∠A), and ∠YAC. The sum of angles on a straight line is 180 degrees. So, ∠XAB + ∠BAC + ∠YAC = 180°. Step 5: Substitute and conclude. Substitute the equalities from Step 3 into the equation from Step 4: ∠B + ∠A + ∠C = 180°. Thus, the sum of the interior angles of a triangle is 180 degrees.

Accepted Answer · 2026-03-29T16:09:55.091Z

Let's prove that the sum of angles in a triangle is 180 degrees. Step 1: Consider a triangle ABC. Let the angles of the triangle be ∠A, ∠B, and ∠C. Step 2: Draw a line parallel to one side. Draw a line XY passing through vertex A such that XY is parallel to the side BC. X-------A-------Y / \ / \ / \ / \ B-----C Step 3: Identify alternate interior angles. Since XY is parallel to BC, and AB is a transversal line: The angle ∠XAB is an alternate interior angle to ∠ABC (which is ∠B). Therefore, ∠XAB = ∠B. Similarly, since XY is parallel to BC, and AC is a transversal line: The angle ∠YAC is an alternate interior angle to ∠ACB (which is ∠C). Therefore, ∠YAC = ∠C. Step 4: Sum the angles on the straight line. The angles on the straight line XY at point A are ∠XAB, ∠BAC (which is ∠A), and ∠YAC. The sum of angles on a straight line is 180 degrees. So, ∠XAB + ∠BAC + ∠YAC = 180°. Step 5: Substitute and conclude. Substitute the equalities from Step 3 into the equation from Step 4: ∠B + ∠A + ∠C = 180°. Thus, the sum of the interior angles of a triangle is 180 degrees.