10.3 Arcs and Chords

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10.3 Arcs and Chords 10.4 Inscribed Angles
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Objectives

• Recognize and use relationships between arcs,
  chords, and diameters
• Find measures of inscribed angles
• Find measures of angles of inscribed polygons

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Arcs and Chords

• Theorem 10.2 In a or ? s, two minor arcs
  are ? iff their corresponding chords are ?.

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Diameters and Chords

• Theorem 10.3 In a , if a diameter (or radius)
  is - to a chord, then it bisects the chord and
  its arc.

If JK - LM, then MO ? LO and arc LK ? arc MK.
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More about Chords

• Theorem 10.4 In a or in ? s, two chords
  are ? iff they are equidistant from the center.

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Inscribed Angles

• An inscribed angle is an angle that has its
  vertex on the circle and its sides are chords of
  the circle.

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Inscribed Angles

• Theorem 10.5 (Inscribed Angle Theorem)The
  measure of an inscribed angle equals ½ the
  measure of its intercepted arc (or the measure of
  the intercepted arc is twice the measure of the
  inscribed angle).

m?ACB ½m or 2 m?ACB
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Example 1
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Example 1
Arc Addition Theorem
Simplify.
Subtract 168 from each side.
Divide each side by 2.
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Example 1
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Example 1
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Your Turn
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Inscribed Angles

• Theorem 10.6If two inscribed ?s intercept ?
  arcs or the same arc, then the ?s are ?.

m?DAC ? m?CBD
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Example 2
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Example 2
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Your Turn
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Your Turn
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Angles of Inscribed Polygons

• Theorem 10.7If an inscribed ? intercepts a
  semicircle, then the ? is a right ?.
• i.e. If AC is a diameter of , then the m?ABC
  90.

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Angles of Inscribed Polygons

• Theorem 10.8If a quadrilateral is inscribed in
  a , then its opposite ?s are supplementary.
• i.e. Quadrilateral ABCD is inscribed in O,
  thus ?A and ?C are supplementary and ?B and ?D
  are supplementary.

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Example 3
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Example 3
Angle Sum Theorem
Simplify.
Subtract 105 from each side.
Divide each side by 3.
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Example 3
Given
Given
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Your Turn
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Example 4
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Example 4
Inscribed Angle Theorem
Sum of angles in circle 360
Subtract 174 from each side.
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Example 4
Inscribed Angle Theorem
Substitution
Divide each side by 2.
Inscribed Angle Theorem
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Example 4
Sum of angles in circle 360
Subtract 204 from each side.
Inscribed Angle Theorem
Divide each side by 2.
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Your Turn
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Assignment

• Geometry Pg. 540 11 29 Pg. 549 8 10, 13
  16, 18, 22 - 25
• Pre-AP Geometry Pg. 540 11 33 Pg. 549 8
  10, 13 30