4.1.2 Angle Relationships in Triangles (continued)

A right triangle is given in this example. One acute angle’s measure is given to be 15.3°. The measure of the second acute angle can be found in two ways. Both methods are explained below, but Method 1 is used in the example to the left. Method 1 Use the Triangle Sum Theorem to find the measure of the unknown angle. By the Triangle Sum Theorem, the sum of the measure of the unknown angle (represented here by ?°), 15.3°, and 90° must be equal to 180°. Use this fact to write an equation and then solve for ?°. Method 2 Use the fact that the sum of the acute angles of a right triangle is 90° to find the measure of the unknown angle. So, the sum of ?° and 15.3° must be 90°. ?° + 15.3° = 90° It follows that ?° = 90° − 15.3° = 74.7°. So, the measure of the other acute angle is 74.7°.

Example 3 Applying the Exterior Angle Theorem The angles of a triangle are classified in three ways in the theorem below: exterior angles, interior angles, and remote interior angles. A triangle’s interior angles are just the three angles inside the triangle. In the figure below, ∠ 1, ∠ 2, and ∠ 3 are interior angles. An exterior angle is formed when one of the triangle’s sides is extended beyond the vertex. The sides of the exterior angle are the extension and the adjacent side of the triangle. Here, ∠ 4 is an exterior angle. Each exterior angle has two remote interior angles, where each remote interior angle is an interior angle that is not adjacent to the exterior angle. Here, ∠ 1 and ∠ 2 are the remote interior angles to ∠ 4.

The Exterior Angle Theorem relates the measure of an exterior angle to the measures of its remote interior angles. If any exterior angle is drawn on a triangle, its measure is equal to the sum of the measures of its remote interior angles.

In this figure, ∠ IJK is an exterior angle because its sides are formed by one side of the triangle and an extension of the adjacent side. The remote interior angles to ∠ IJK in △ HIJ are ∠ H and ∠ I . So, by the Exterior Angle Theorem, m ∠ IJK must be equal to the sum of m ∠ H and m ∠ I . Use this fact to write an equation. Substitute the expressions given in the figure for m ∠ H and m ∠ I and the given measure of ∠ IJK into the equation and solve for x . Then, substitute the value of x into the expression given for m ∠ H and simplify to find m ∠ H .

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