How can you prove that the angles in any triangle sum up to 180°?

To prove that the angles in any triangle sum up to 180°, one effective method is through the exterior angle theorem. This theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

To illustrate this, consider a triangle labeled ABC. If you extend one side, say BC, to form an exterior angle at vertex C, the exterior angle created is equal to the sum of the angles A and B inside the triangle. Therefore, if you set the measure of angle C plus the measure of angle A plus the measure of angle B together with the principle that an exterior angle and the adjacent interior angle (angle C) sum up to 180° (which is a straight line), you derive that:

Angle A + Angle B + Angle C = 180°.

This foundational property proves that the sum of internal angles of any triangle consistently totals 180°, validating the geometry involved in triangle formation.

This method provides a clear geometric visual and a logical flow to the proof, reinforcing the understanding of angle relationships in triangles.