Proof that the sum of triangle is 180 0
ScanSolve AI Answer
Step-by-step solution
5 stepsLet's prove that the sum of angles in a triangle is 180 degrees.
Consider a triangle ABC.
Let the angles of the triangle be ∠A, ∠B, and ∠C.
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
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.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°.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.
Still stuck on this one?
Ask a follow-up, and the answer starts from this question.