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

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

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

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

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

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

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

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

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

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 (268 - 92) = x 200 – x = 144 ½ (176) = x – x = -56 Mrs. McConaughy Geometry: Circles

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

Final Checks for Understanding Mrs. McConaughy Geometry: Circles

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