Download Other Angle Relationships in Circles

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 an inscribed
arc.
angle equals _____________
_______________________.
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.
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 ___ a
circle, then the measure
of each angle formed is
___________________
___________________.
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___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 ________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.
Mrs. McConaughy
Geometry: Circles
8
Example 2
Lines Intersecting INSIDE a
Circle: Finding the Measure
Angles Formed by Two
Chords
Find x.
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.
Mrs. McConaughy
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