Central Angles and Arc Measures
A central angle has its vertex at the center of a circle with its sides formed by radii. The sum of all central angles in any circle is always 360°. When you see a central angle, remember that it creates an arc on the circle.
The measure of an arc equals the measure of its central angle. For example, if angle ABC is 84°, then the arc AC it creates also measures 84°. If you need to find a complementary arc (the rest of the circle), just subtract from 360° - like arc ADC = 360° - 84° = 276°.
Working with central angles means recognizing patterns. When given angle measures, you can determine arc measures directly, and vice versa. For multi-part problems, remember to account for the entire 360° of the circle when calculating unknown angles.
Quick Tip: When solving for an unknown variable in central angle problems, set up an equation based on the fact that all angles in a circle sum to 360°. This gives you a clear starting point for any problem!
For example, in problems with algebraic expressions like ° and °, you'll need to set up equations using the 360° total. Solving these equations reveals the value of x, which then helps you find all remaining angle and arc measures.
