Section 6.2 More Angle Measures in a Circle Tangent is a line that intersects a circle at exactly one point; the point of intersection is the point of contact, or point of tangency. Secant is a line (or segment or ray) that intersects a circle at exactly two points. 1/15/2019 Section 6.2 Nack

Polygons Inscribed in a Circle A polygon is inscribed in a circle if its vertices are points on the circle and its sides are chords of the circle. Equivalent statement: A circle is circumscribed about the polygon Cyclic polygon: the polygon inscribed in the circle. Theorem 6.2.1: If a quadrilateral is inscribed in a circle, the opposite angles are supplementary. 1/15/2019 Section 6.2 Nack

Polygons Circumscribed about a circle A polygon is circumscribed about a circle if all sides of the polygon are line segments tangent to the circle; the circle is said to be inscribed in the polygon. 1/15/2019 Section 6.2 Nack

Measure of an Angle Formed by Two Chords that Intersect Within a Circle Theorem 6.2.2: The measure of an angle formed by two chords that intersect within a circle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 1. Proof: Draw CB. Two points determine a line 2. m1 = m2 + m3 1 is an exterior  of ΔCBE. 3. m2 = ½ mDB 2 and 3 are inscribed m3 = ½ mAC  ‘s of O. 4. m1 = ½ (mDB + mAC) Substitution 1/15/2019 Section 6.2 Nack

Theorems with Tangents and Secants Theorem 6.2.3: The radius or diameter drawn to a tangent at the point of tangency is perpendicular to the tangent at that point. See Fig. 6.29 (a,b,c) Ex. 2 p. 289 Corollary 6.2.4: The measure of an angle formed by a tangent and a chord drawn to the point of tangency is one-half the measure of the intercepted arc. Example 3 p. 289 1/15/2019 Section 6.2 Nack

Measures of Angles formed by Secants. Theorem 6.2.5: The measure of an angle formed when two secants intersect at a point outside the circle is one-half the difference of the measures of the two intersecting arcs. Given: Secants AC and DC Prove: mC = ½(m AD – m BE) Draw BD to form ΔBCD m1 = mC + mD Exterior angles of ΔBCD mC = m1 - mD Addition Property m1 = ½m AD Inscribed Angles mD= ½ m BE mC= ½m AD - ½ m BE Substitution mC= ½(m AD – m BE) Distributive Law 1/15/2019 Section 6.2 Nack

Theorem 6.2.6: If an angle is formed by a secant and a tangent that intersect in the exterior of a circle, then the measure of the angle is one-half the difference of the measure of its intercepted arcs. Theorem 6.2.7: If an angle is formed by two intersecting tangents, then the measure of the angle is one-half the difference of the measures of the intercepted arcs. 1/15/2019 Section 6.2 Nack

Parallel Chords Theorem 6.2.8: If two parallel lines intersect in a circle, the intercepted arcs between these lines are congruent. AC  BD A B C D 1/15/2019 Section 6.2 Nack

Summary of Methods for Measuring Angles Related to a Circle. Location of the Vertex of the Angle Center of the Circle In the interior of the circle On the Circle In the exterior of the circle Example 6 p. 292 Rule for Measuring the Angle The measure of the intercepted arc. One-half the sum of the measures of the intercepted arcs. One-half the measure of the intercepted arc. One-half the difference of the measures of the two intercepted arcs. 1/15/2019 Section 6.2 Nack