In this explanation will be discussed how to prove the theorem of the sum of the angles of a triangle. Pay attention to the explanation below:
Theorem:
"The sum of the angles of a triangle is 180"
Let the straight line BC
Let C be a point between B and D
Let E be a point such that A and E are on the same side of line BD
Suppose CE to be the line through C parallel to line BA
Playfair's Axiom:
"Through a given point there can be only one straight line parallel to a given straight line"
Considering to the playfair's axiom, line CE is the only one straight line that parallel to line BA
Theorem 1
"If two parallel line are cut by a tranversal, the alternate interior angles are equal"
Since AB and CE are parallel and AC as tranversal, by theorem 1 we obtain:
the measure of ACE angle equal to the measure of BAC angle
Theorem 2
"If two parallel line are cut by a tranversal, the corresponding angles are equal"
Since AB and CE are parallel and BD as tranversal, by theorem 2 we obtain:
the measure of ECD angle equal to the measure of ABC angle
Corollary:
"For any triangle, the measure of an exterior angle is the sum of the measures of its two remote interior angles."
The measure of ACD angle is the sum of the measure of CAB angle and the measure of ABC angle.
As we know, the sum all of angles in straight line is 180
So we obtain,
The sum of the measure of BCA angle and the measure of ABC angle equal to 180
Since the sum of the measure of CAB angle and the measure of ABC angle equal to the measure of ACD angle
Thus,
The sum of the measure of BCA, the measure of CAB angle and the measure of ABC angle equal to 180
which was to be proved
