finding arc measures\na satellite views the earth at an angle of $20^\\circ$. what is the arc measure, $x$…

finding arc measures\na satellite views the earth at an angle of $20^\\circ$. what is the arc measure, $x$, that the satellite can see?\n$40^\\circ$\n$80^\\circ$\n$160^\\circ$\n$320^\\circ$\nearth\nsatellite

finding arc measures\na satellite views the earth at an angle of $20^\\circ$. what is the arc measure, $x$, that the satellite can see?\n$40^\\circ$\n$80^\\circ$\n$160^\\circ$\n$320^\\circ$\nearth\nsatellite

Answer

Explanation:

Step1: Recall external circle angle formula

The measure of an angle formed outside a circle by two tangents is half the difference of the measures of the intercepted arcs. The total circumference of a circle is $360^\circ$, so the intercepted arcs are $x$ and $360^\circ - x$.

Step2: Set up the equation

Let the external angle be $20^\circ$. Apply the formula: $$20^\circ = \frac{1}{2}\left((360^\circ - x) - x\right)$$

Step3: Simplify and solve for $x$

First simplify the right-hand side: $$20^\circ = \frac{1}{2}(360^\circ - 2x)$$ Multiply both sides by 2: $$40^\circ = 360^\circ - 2x$$ Rearrange to solve for $x$: $$2x = 360^\circ - 40^\circ$$ $$2x = 320^\circ$$ $$x = 160^\circ$$

Answer:

160°