The exterior angle at a vertex of a triangle is equal to the sum of the remote interior angles.
We know that the sum of the angles in a triangle is 180°. So now it is simple to prove a corollary theorem.
Here we will prove that the measure of an exterior angle at a vertex of a triangle is equal the to the sum of the measures of the interior angles at the other two vertices of the triangle (called the remote interior angles).
(1) m∠1 + m∠4 = 180° // straight line measures 180°
(2) m∠2 + m∠3+ m∠4 = 180° //sum of the interior angles in a triangle
(3) m∠1 + m∠4 = m∠2 + m∠3+ m∠4 //using (1) and (2) and transitive property of equality, both equal 180°
(4) m∠1 = m∠2 + m∠3 // subtraction property of equality (subtracted m∠4 from both sides)
Strategy for solving this problem:
Here we combine our knowledge of two sets of angles that sum up to 180° . A linear pair of angles (that make up a straight line) , and the interior angles of a triangle.
Since both sets add up to the same total, and one angle is common to both (∠4), the others must be equal.