Central Angles

Acentral angleis an angle with its vertex at the center of a circle, with its sides containing tworadiiof the circle.

In the figure above, $∠ P Z Q, ∠ Q Z R$ , and $∠ R Z P$ are central angles.

Sum of Central Angles:The sum of the measures of the central angles of a circle with no points in common is $360 °$ .

For example, in the figure above,

$m ∠ P Z R + m ∠ R Z Q + m ∠ Q Z P = 360 °$ .

Example:

Find the value of $x$ .

The sum of the measures of the central angles of a circle with no interior points in common is $360 °$ . So,

$\begin{array}{l} m ∠ M O N + m ∠ N O P + m ∠ P O M = 360 ° \\ 50 ° + 90 ° + x = 360 ° \end{array}$

Simplify.

$140 ° + x = 360 °$

Subtract $140 °$ from each side.

$x = 220 °$