Download By the Interior Angle Theorem, the measure of an interior angle in a

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°.