Prove that 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 to the sum of the measures of the interior angles at the other two vertices of the triangle (called the remote interior angles).
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.
(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)