Question

b. determine the measure of each unknown angle. list the side lengths from shortest to longest for each diagram. explain your reasoning.
1.
2.

Explanation:

Step1: Find unknown angles

In a triangle, the sum of interior angles is 180°.
For the first triangle:
Let's find the unknown - angle calculations are correct. The sum of the given angles \(47^{\circ}+22^{\circ}=69^{\circ}\), and the third - angle is \(180^{\circ}-69^{\circ}=111^{\circ}\).
For the second triangle: The sum of the given angles \(55^{\circ}+59^{\circ}=114^{\circ}\), and the third - angle is \(180^{\circ}-114^{\circ}=66^{\circ}\).

Step2: Relate angles and side lengths

The side opposite the smallest angle is the shortest, and the side opposite the largest angle is the longest.
For the first triangle:
The angles are \(22^{\circ}\), \(47^{\circ}\), and \(111^{\circ}\). The side opposite \(22^{\circ}\) is \(b\), the side opposite \(47^{\circ}\) is \(a\), and the side opposite \(111^{\circ}\) is \(c\). So the order of side lengths from shortest to longest is \(b\), \(a\), \(c\).
For the second triangle:
The angles are \(36^{\circ}\), \(55^{\circ}\), and \(59^{\circ}\). The side opposite \(36^{\circ}\) is \(s\), the side opposite \(55^{\circ}\) is \(t\), and the side opposite \(59^{\circ}\) is \(r\). So the order of side lengths from shortest to longest is \(s\), \(t\), \(r\).

Answer:

1. Unknown angles: \(111^{\circ}\). Side - length order: \(b\), \(a\), \(c\).
2. Unknown angles: \(66^{\circ}\). Side - length order: \(s\), \(t\), \(r\).