Question

check your understanding

1. find the measures of the indicated angles.

b = 25°
65°
b
45°
64°

Explanation:

Step1: Analyze left - hand side figure

In the left - hand side figure, we know that the sum of angles in a right - angled corner is 90°. Given one angle is 65°, and we want to find angle \(b\). We use the formula \(b = 90^{\circ}-65^{\circ}\).
\[b=90^{\circ}-65^{\circ}=25^{\circ}\]

Step2: Analyze right - hand side figure

In the right - hand side figure, we know that vertical angles are equal. Let the unknown angles be \(x\). We use the fact that the sum of angles around a point is 360°. Also, we know that vertical angles are equal. So, if one of the non - given angles is \(x\), and we have two pairs of vertical angles. One pair has an angle of 64° and the other has an angle of 45°. We can find \(x\) as follows:
First, we know that \(2x+64^{\circ}+45^{\circ}=180^{\circ}\) (since the angles on one side of a straight line through the intersection point sum to 180°).
\[2x=180^{\circ}-(64^{\circ} + 45^{\circ})\]
\[2x=180^{\circ}-109^{\circ}\]
\[2x = 71^{\circ}\]
\[x=\frac{71^{\circ}}{2}=35.5^{\circ}\]

Answer:

For the left - hand side figure, \(b = 25^{\circ}\). For the right - hand side figure, the measure of each of the indicated angles is \(35.5^{\circ}\)