Here are three important theorems about triangles: sum of angles of a triangle, the measure of the exterior angle, and the base angles of an isosceles triangle. Sample problems where these are applied are also provided. The problems involved the concept of ratio.
Theorem: The Sum of the Angles of a Triangle
The sum of the measures of the angles of a triangle is 180°.
In the figure, α+β+δ = 180°.
Problem 1: If the measure of the angles in a triangle is in the ratio 1:2:3, what is the measure of the biggest angle?
Solution: Let x be the smallest angle. It follows that the next bigger angle has measure 2x and the biggest angle has measure 3x. Their sum is x + 2x + 3x = 180°. This simplifies to 6x = 180° and x = 30°. The biggest angle, 3x, is 3 × 30° = 90°.
Theorem: The Exterior Angle of a Triangle
The exterior angle of a triangle is equal in size to the sum of the interior opposite angles.
Let α, β, and δ be the measures of the angles of a triangle. The exterior angle of the triangle at A is equal to β + δ; at B is α + δ ; and at C is α + β.
Problem 2: If the measure of the angles in a triangle is in the ratio 1:2:3, what is the measure of the exterior angle at the second to the biggest?
Solution: From the solution to problem 1, we know that the smallest angle is 30° and the biggest angle is 90°. Therefore the exterior angle at the second biggest angle is 30° + 90°.
Theorem: The Base Angles of an Isosceles Triangle
The base angles of an isosceles triangle are equal.
In the figure, ΔABC is an isosceles triangle. The two equal sides are AB and BC. The base angles A and B have equal measure. That is , α = α. The third angle, angle B in the drawing below, is called the vertex angle.
Problem 3. If the ratio of the measure of the vertex angle to the base angle of an isosceles triangle is 2:1, what is the measure of the vertex angle?
Solution: Since base angles of an isosceles triangle are equal and it is given in the problem that the ratio of the vertex angle to one of the base angles is 2:1, the measure of the angles of the triangle is in the ratio 2:1:1. Let x be the measure of the base angles. This implies that x + x + 2x = 180°. This simplifies to 4x = 180° and x = 45°. Thus, the vertex angle is 2(45) = 90°.
Proofs of theorems will be in the next few posts.