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Difficulty: MediumRadians and Degrees

In a circle with center OO, the central angle AOBAOB has a measure of 4π5\frac{4\pi}{5} radians. Angle BOCBOC is adjacent to angle AOBAOB such that they form a straight line. What is the degree measure of angle BOCBOC?

Answer: 36

Answer

The degree measure of angle BOCBOC is 36.
Angles AOBAOB and BOCBOC form a straight line, which means they are supplementary and their measures add up to 180∘180^\circ (or π\pi radians). Since angle AOBAOB measures 4π5\frac{4\pi}{5} radians, we can convert this measure to degrees first by multiplying by 180∘π\frac{180^\circ}{\pi}: 4π5×180∘π=4×36∘=144∘\frac{4\pi}{5} \times \frac{180^\circ}{\pi} = 4 \times 36^\circ = 144^\circ. To find the degree measure of the supplementary angle BOCBOC, subtract 144∘144^\circ from 180∘180^\circ, giving 180∘−144∘=36∘180^\circ - 144^\circ = 36^\circ. Alternatively, the calculation can be performed in radians first: π−4π5=π5\pi - \frac{4\pi}{5} = \frac{\pi}{5} radians, and then converting π5\frac{\pi}{5} radians to degrees: π5×180∘π=36∘\frac{\pi}{5} \times \frac{180^\circ}{\pi} = 36^\circ.

Step-by-Step Solution

1
Identify the relationship between the two adjacent angles.
The sum of the measures of angles AOBAOB and BOCBOC is π\pi radians or 180∘180^\circ.
Since the adjacent angles AOBAOB and BOCBOC form a straight line, they are supplementary.
2
Calculate the measure of angle BOCBOC in radians.
Angle BOCBOC measures π5\frac{\pi}{5} radians.
Subtract the measure of angle AOBAOB from the straight line measure: π−4π5=π5\pi - \frac{4\pi}{5} = \frac{\pi}{5} radians.
3
Convert the radian measure of angle BOCBOC to degrees.
The degree measure is 36.
Multiply the radian measure by the conversion factor 180∘π\frac{180^\circ}{\pi}: π5×180π=36∘\frac{\pi}{5} \times \frac{180}{\pi} = 36^\circ.

Key Concept

Converting angles from radians to degrees and using the properties of supplementary angles.
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Topics

Radians and Degrees

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