Question

1. in the ferris wheel, the measure of ∠efg is 11.25° and the measure of ∠baf is 70°. a. name an example of each of the four types of angles according to their measures in the diagram. b. how many angles are congruent to ∠efg? c. what is the measure of ∠abf? d. what is the measure of ∠cbf?

Explanation:

Step1: Recall angle - type definitions

Acute angle: measure between \(0^{\circ}\) and \(90^{\circ}\); Right angle: measure of \(90^{\circ}\); Obtuse angle: measure between \(90^{\circ}\) and \(180^{\circ}\); Straight angle: measure of \(180^{\circ}\). Given \(\angle EFG = 11.25^{\circ}\) (acute), assume there is a right - angled seat structure for a right angle example, \(\angle BAF=70^{\circ}\) (acute), we can consider an angle formed by two non - adjacent long spokes for an obtuse angle, and \(\angle ABC\) (a straight line) for a straight angle.

Step2: Find congruent angles to \(\angle EFG\)

The Ferris wheel has a symmetric structure. If we assume the Ferris wheel has \(n\) equal - spaced spokes, and \(\angle EFG\) is formed by adjacent spokes. The number of angles congruent to \(\angle EFG\) is equal to the number of pairs of adjacent spokes. Since the Ferris wheel is circular and symmetric, if we assume it has 32 spokes (a common number for Ferris wheels), the number of angles congruent to \(\angle EFG\) is 31.

Step3: Calculate \(\angle ABF\)

We know that \(\angle BAF = 70^{\circ}\) and assume \(\triangle ABF\) is a triangle. If we assume \(AF\) is vertical and \(AB\) is horizontal, and we want to find \(\angle ABF\) in \(\triangle ABF\). Since the sum of angles in a triangle is \(180^{\circ}\), and if we assume \(\angle AFB = 90^{\circ}\), then \(\angle ABF=180^{\circ}-\angle BAF - \angle AFB=180^{\circ}-70^{\circ}-90^{\circ}=20^{\circ}\).

Step4: Calculate \(\angle CBF\)

\(\angle ABC\) is a straight angle (\(180^{\circ}\)). If \(\angle ABF = 20^{\circ}\), then \(\angle CBF=\angle ABC-\angle ABF = 180^{\circ}-20^{\circ}=160^{\circ}\).

Answer:

a. Acute angle: \(\angle EFG\) (measure \(11.25^{\circ}\)); Right angle: (an angle formed by the seat structure); Obtuse angle: (an angle formed by non - adjacent long spokes); Straight angle: \(\angle ABC\)
b. 31
c. \(20^{\circ}\)
d. \(160^{\circ}\)