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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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section 6 2 more angle measures in a circle
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.

Section 6.2 Nack

polygons inscribed in a circle
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.

Section 6.2 Nack

polygons circumscribed about a circle
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.

Section 6.2 Nack

measure of an angle formed by two chords that intersect within a circle
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

Section 6.2 Nack

theorems with tangents and secants
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

Section 6.2 Nack

measures of angles formed by secants
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

Section 6.2 Nack

slide7
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.

Section 6.2 Nack

parallel chords
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

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.

Summary of Methods for Measuring Angles Related to a Circle.

Section 6.2 Nack