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.
ScanSolve AI Answer
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.