Section 10.4 Other Angle Relationships in Circles

Intersection ON the circle (theorem 10.12) If a tangent and a chord intersect at a point on the circle, then the measure of each angle formed is one half the measure of the intercepted arc m  1 = ½ m AB m  2 = ½ m BCA Ex: 120° m  1 = ½ (120 °)=60°

Intersection INSIDE the circle (theorem 10.13) If two chords intersect inside the circle, then each angle measures one half the sum of the measures of the intercepted arcs m  1 = ½ (m CD + m AB) Ex: 30° 100° m  1 = ½ (100 ° + 30°) = 65 °

Intersection OUTSIDE the circle (theorem 10.14) If a tangent and a secant, two tangents, or two secants intersect outside the circle, then each angle measures one half the difference of the intercepted arcs.

Tangent and Secant 1 A B C D m  1 = ½ (m BC – m AC) 200° 60° m  1 = ½ (200 ° - 60 °)=70 °

Two Tangents 2 P R Q m  2 = ½ (m PQR – m PR) 280° 80° m  2 = ½ (280 ° - 80 °)=100 °

Two Secants 3 W Z X Y m  3 = ½ (m XY – m WZ) 100° 30° m  3 = ½ (100 ° - 30 °)=35 °

Practice Problems Intersection on the circle: Angle = ½ arc Intersection inside the circle: Angle = (arc + arc) 2 Intersection outside the circle: Angle = large arc – small arc 2