Question

lving a real-world problem
aaron is standing at point c, watching his friends on a ferris wheel. he knows that he is looking up at a $57^\circ$ angle and the measure of arc bd is $80^\circ$. what is the measure of arc aed?
$\boldsymbol{\square}^\circ$

Explanation:

Step1: Use secant-secant angle theorem

The measure of an angle formed by two secants outside a circle is half the difference of the measures of the intercepted arcs. So:
$$57^\circ = \frac{1}{2} \times (\text{arc } AED - \text{arc } BD)$$

Step2: Substitute arc BD value

Replace arc BD with $80^\circ$:
$$57^\circ = \frac{1}{2} \times (\text{arc } AED - 80^\circ)$$

Step3: Solve for arc AED

Multiply both sides by 2, then add $80^\circ$:
$$\text{arc } AED = (57^\circ \times 2) + 80^\circ = 114^\circ + 80^\circ$$

Answer:

$194^\circ$