Question

use the figure for exercises 13 - 15. find the measure of the given angle. 13. a 14. b 15. c envision® geometry • teaching resources

Explanation:

Step1: Use angle - sum property of a triangle

The sum of angles in a triangle is 180°. For the left - hand triangle, we know one exterior angle is 112°. The non - adjacent interior angles are 33° and \(a\). An exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(112^{\circ}=33^{\circ}+a\).

Step2: Solve for \(a\)

Subtract 33° from both sides of the equation \(112^{\circ}=33^{\circ}+a\). We get \(a = 112^{\circ}-33^{\circ}=79^{\circ}\).

Step3: Find \(b\)

The angle adjacent to 112° forms a linear pair with 112°. So, this angle is \(180^{\circ}-112^{\circ}=68^{\circ}\). In the middle triangle, we know two angles: 68° and 43°. Using the angle - sum property of a triangle (\(180^{\circ}\) for the sum of interior angles of a triangle), we have \(b=180^{\circ}-(68^{\circ} + 43^{\circ})=180^{\circ}-111^{\circ}=69^{\circ}\).

Step4: Find \(c\)

In the right - hand triangle, we know one angle is \(71^{\circ}\) and another is \(b = 69^{\circ}\). Using the angle - sum property of a triangle, \(c=180^{\circ}-(71^{\circ}+69^{\circ})=180^{\circ}-140^{\circ}=40^{\circ}\).

Answer:

1. \(a = 79^{\circ}\)
2. \(b = 69^{\circ}\)
3. \(c = 40^{\circ}\)