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Mrs. McConaughy Geometry: Circles 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 measure of an inscribed angle equals ____________________________________.
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.
one-half the measure of its intercepted arc.
The following theorems will help to determinethe measures of angles formed by lines which intersect on, inside or outside a circle.
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 determinethe measures of angles formed by lines which intersect inside or outside a circle.
Mrs. McConaughy Geometry: Circles 4
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 = _____
Measures of Angles Formed by Lines Intersecting ON a Circle = ½ the measure of the intercepted arc.
Mrs. McConaughy Geometry: Circles 5
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.
Measure of angle 1 = _____
Measure of angle 2 = ______
Measures of Angles Formed by Chords Intersecting INSIDE a Circle = ½ the SUM of the Intercepted Arcs
Mrs. McConaughy Geometry: Circles 6
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.
Measures of Angles Formed by Secants and/or Tangents Intersecting OUTSIDE a Circle = ½ the DIFFERENCE of the Intercepted Arcs
Mrs. McConaughy Geometry: Circles 7
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
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.
m < 1 = ½ intercepted arc
130 = ½ intercepted arc
260m < 1 = ½ (150)
m <1 = _____75
260 = intercepted arc
Measures of Angles Formed by Lines Intersecting ON a Circle = ½ the measure of the 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 (174 + 106) = 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.
56 88
½ (200 – x) = 72
200 – x = 144
– x = -56
360-92= 268
½ (268 - 92) = x
½ (176) = x
Measures of Angles Formed by Secants and/or Tangents Intersecting OUTSIDE a Circle = ½ the DIFFERENCE of the Intercepted Arcs
Mrs. McConaughy Geometry: Circles 11
In summary:
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.
The measure of an angle formed equals ½ the difference 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
