10.4%20Inscribed%20Angles

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10.4 Inscribed Angles
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Objectives

• Find measures of inscribed angles
• Find measures of angles of inscribed polygons

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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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Example 3
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Example 3
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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 4
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Example 4
Angle Sum Theorem
Simplify.
Subtract 105 from each side.
Divide each side by 3.
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Example 4
Given
Given
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Your Turn
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Example 5
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Example 5
Inscribed Angle Theorem
Sum of angles in circle 360
Subtract 174 from each side.
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Example 5
Inscribed Angle Theorem
Substitution
Divide each side by 2.
Inscribed Angle Theorem
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Example 5
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. 549 8 10, 13 16, 18 20
• Pre-AP Geometry Pg. 549 8 10, 13 20