Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Interior and Exterior Angles in Circles Use the Interior Angle Theorem A line that intersects a circle at two points is called a secant of a circle. In the figure below, line AB is a secant of circle P. If two non-parallel secants intersect, the intersection point can lie in the interior of the circle or the exterior of the circle. If two secants intersect in the interior of a circle, then interior angles are formed. The Interior Angle Theorem states that the measure of an interior angle in a circle is equal to half the sum of the measures of the interior angle's intercepted arc and the intercepted arc of the interior angle's vertical angle. In the figure of circle C below, the measure of angle AHG is equal to half the sum of the measures of arcs AG and FB. Example: In circle H, the measure of arc JL is 92° and the measure of arc MK is 35°. Find the measure of angle MNK. By the Interior Angle Theorem, the measure of an interior angle in a circle is equal to half the sum of the measures of the arcs intercepted by the angle and its vertical angle. You are given that the measure of arc JL is 92° and the measure of arc MK is 35°. You can use this information to write and solve an equation for the measure of angle MNK. m∠MNK = 12(measure of arc MK + measure of arc JL) m∠MNK = 12(35° + 92°) Substitute the given arc measures. m∠MNK = 12(127°) Add. m∠MNK = 63.5° Multiply. So, the measure of angle MNK is 63.5°. Use the Exterior Angle Theorem If two secants intersect in the exterior of a circle, then exterior angles are formed. The Exterior Angle Theorem states that the measure of an exterior angle of a circle is equal to half the difference of the measures of the two intercepted arcs. In the figure of circle O below, the measure of angle BFD is equal to half the difference of the measures of arcs BD and AC. Example: In circle T, the measure of angle QVS is 58° and the measure of arc PR is 21°. Find the measure of arc QS. By the Exterior Angle Theorem, the measure of an exterior angle of a circle is equal to half the difference of the measures of the two intercepted arcs. You are given that the measure of angle QVS is 58° and the measure of arc PR is 21°. Let x represent the measure of arc QS. You can use this information to write an equation. Then solve the equation for x to find the measure of arc QS. 12(measure of arc QS - measure of m∠QVS = arc PR) 58° = 12(x 21°) Substitute the given measures. 116° = x - 21° Multiply both sides by 2. 137° = x Add 21° to both sides. So, the measure of arc QS is 137°.