Other Angle
Relationships in
Circles
In this lesson, you will use
angles formed by lines that
intersect a circle to solve
problems
Mrs. McConaughy
Geometry: Circles
1
We already know how
to find the measures
of several angles and
their intercepted
arcs.
Recall,
The measure of a central
angle equals _____________
the measure of
_______________________.
its intercepted arc.
The following theorems will help to determine
Themeasures
measure of
of angles
an inscribed
the
formed by lines which
one-half the
angle equals
intersect
on,_____________
inside or outside a circle.
measure
of its intercepted
_______________________.
arc.
Mrs. McConaughy
Geometry: Circles
2
Lines Intersecting INSIDE,
OUTSIDE, or ON a Circle
If two lines intersect a circle, there are
three places where the lines can
intersect.
The following theorems will help to determine
the measures of angles formed by lines which
intersect inside or outside a circle.
Mrs. McConaughy
Geometry: Circles
3
Measures of Angles Formed by Lines Intersecting
ON a Circle = ½ the measure of the intercepted arc.
THEOREM
If a tangent and a chord
intersect at a point on a
circle, then the measure
of each angle formed is
½
the measure of the
___________________
intercepted
arc
___________________.
Measure of angle 1 = _____
Measure of angle 2 = _____
Mrs. McConaughy
Geometry: Circles
4
Measures of Angles Formed by Chords Intersecting
INSIDE a Circle = ½ the SUM of the Intercepted Arcs
THEOREM
If two chords intersect in
the interior of a circle,
then the measure of each
angle formed is one-half
the sum of the measures
of the arcs intercepted
by the angle and its
vertical angle.
Mrs. McConaughy
Geometry: Circles
Measure of angle 1 = _____
Measure of angle 2 = ______
5
Measures of Angles Formed by Secants and/or
Tangents Intersecting OUTSIDE a Circle = ½ the
DIFFERENCE of the Intercepted Arcs
THEOREM
If a secant and a tangent, two tangents, or two
secants intersect in the exterior of a circle,
then the measure of the angle formed is
one-half the difference of the measures of
the intercepted arcs.
Mrs. McConaughy
Geometry: Circles
6
Measures of Angles Formed by Secants and/or
Tangents Intersecting OUTSIDE a Circle
Case I:
Case II: Two
Tangent and a Tangents
Secant
Mrs. McConaughy
Geometry: Circles
Case III: Two
Secants
7
Measures of Angles Formed by Lines Intersecting
Lines
Intersecting
ON
a
Example
1
ON a Circle = ½ the measure of the
intercepted arc.Circle: Finding Angle and Arc
Measures
Line m is tangent to the circle. Find the
measure of the red angle or arc.
m < 1 = ½ intercepted arc
m < 1 = ½ (150)
m <1 = _____
75
Mrs. McConaughy
260
130 = ½ intercepted arc
260 = intercepted arc
Geometry: Circles
8
Example 2
Lines Intersecting INSIDE a
Circle: Finding the Measure
Angles Formed by Two
Chords
Find x.
½ (174 + 106) = X
½ (280) = X
140 = X
140
Measures of Angles Formed by Chords Intersecting
Mrs. McConaughy
INSIDE
a Circle = ½ theGeometry:
SUM Circles
of the Intercepted Arcs
9
Example 3 LINES INTERSECTING OUTSIDE A
Measures ofCIRCLE:
Angles Formed
by Secants
and/or of an
Finding
the Measure
Tangents Intersecting OUTSIDE a Circle = ½ the
Angle Formed by Secants and/or
DIFFERENCE of the Intercepted Arcs
Tangents
Find the value of x.
56
88
360-92= 268 
½ (200 – x) = 72
200 – x = 144
– x = -56
Mrs. McConaughy
½ (268 - 92) = x
½ (176) = x
Geometry: Circles
10
In summary:
The measure of
an angle formed
The measure The
of measure of an angle formed
equals ½ the
an angle formed
equals ½ the sum of the measures
difference of the
equals ½ its of the arcs intercepted by the
measures of the
intercepted arc.
angle and its vertical angle.
arcs intercepted
by the angle and
its vertical angle.
Mrs. McConaughy
Geometry: Circles
11
Final Checks for Understanding
Mrs. McConaughy
Geometry: Circles
12
Homework Assignment
Angle Relationships in Triangles WS
Mrs. McConaughy
Geometry: Circles
13