1. 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.
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2. 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.
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3. 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.
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4. 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
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5. 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 (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
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6. 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
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7. 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.
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8. 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
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9. 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.
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