1. 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

2. The following theorems will help to determine
We already know how to find the measures of several angles and their intercepted arcs.
Recall,
The measure of a central angle equals _____________ _______________________.
The following theorems will help to determine
the measures of angles formed by lines which
intersect on, inside or outside a circle.
the measure of its intercepted arc.
The measure of an inscribed angle equals _____________ _______________________.
one-half the measure of its intercepted arc.
Mrs. McConaughy
Geometry: Circles

3. 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

4. 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

5. 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
Measure of angle 1 = _____
Measure of angle 2 = ______
one-half the sum of the measures of the arcs intercepted
by the angle and its vertical angle.
Mrs. McConaughy
Geometry: Circles

6. 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

7. Case I: Tangent and a Secant
Measures of Angles Formed by Secants and/or Tangents Intersecting OUTSIDE a Circle
Case I: Tangent and a Secant
Case II: Two Tangents
Case III: Two Secants
Mrs. McConaughy
Geometry: Circles

8. Measures of Angles Formed by Lines Intersecting
ON a Circle = ½ the measure of the
intercepted arc.
Example 1
Lines Intersecting ON a Circle: Finding Angle and Arc Measures
Line m is tangent to the circle. Find the measure of the red angle or arc.
260
m < 1 = ½ intercepted arc
m < 1 = ½ (150)
m <1 = _____ 75
130 = ½ intercepted arc
260 = intercepted arc
Mrs. McConaughy
Geometry: Circles

9. Example 2
Lines Intersecting INSIDE a Circle: Finding the Measure Angles Formed by Two Chords
Find x.
½ ( ) = X
½ (280) = X
140 = X
140
Measures of Angles Formed by Chords Intersecting
INSIDE a Circle = ½ the SUM of the Intercepted Arcs
Mrs. McConaughy
Geometry: Circles

10. Example 3
LINES INTERSECTING OUTSIDE A CIRCLE: Finding the Measure of an Angle Formed by Secants and/or Tangents
Find the value of x.
Measures of Angles Formed by Secants and/or
Tangents Intersecting OUTSIDE a Circle = ½ the
DIFFERENCE of the Intercepted Arcs 56 88
360-92= 268 
½ (200 – x) = 72
½ ( ) = x
200 – x = 144
½ (176) = x
– x = -56
Mrs. McConaughy
Geometry: Circles

11. In summary:
The measure of an angle formed equals ½ the difference of the measures of the arcs intercepted by the angle and its vertical angle.
The measure of an angle formed equals ½ its intercepted arc.
The measure of an angle formed equals ½ the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Mrs. McConaughy
Geometry: Circles

12. Final Checks for Understanding
Mrs. McConaughy
Geometry: Circles

13. Homework Assignment Angle Relationships in Triangles WS
Mrs. McConaughy
Geometry: Circles