10.4 Other Angle Relationships in Circles

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10.4 Other Angle Relationships in Circles

• Geometry
• Mrs. Spitz
• Spring 2005

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Objectives/Assignment

• Use angles formed by tangents and chords to solve
  problems in geometry.
• Use angles formed by lines that intersect a
  circle to solve problems.
• Assignment pp. 624-625 2-35

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Using Tangents and Chords

• You know that measure of an angle inscribed in a
  circle is half the measure of its intercepted
  arc. This is true even if one side of the angle
  is tangent to the circle.

m?ADB ½m
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Theorem 10.12

• If a tangent and a chord intersect at a point on
  a circle, then the measure of each angle formed
  is one half the measure of its intercepted arc.

m?1 ½m
m?2 ½m
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Ex. 1 Finding Angle and Arc Measures

• Line m is tangent to the circle. Find the
  measure of the red angle or arc.
• Solution
• m?1 ½
• m?1 ½ (150)
• m?1 75

150
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Ex. 1 Finding Angle and Arc Measures

• Line m is tangent to the circle. Find the
  measure of the red angle or arc.
• Solution
• m 2(130)
• m 260

130
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Ex. 2 Finding an Angle Measure

• In the diagram below,
• is tangent to the circle. Find m?CBD
• Solution
• m?CBD ½ m
• 5x ½(9x 20)
• 10x 9x 20
• x 20
• ? m?CBD 5(20) 100

(9x 20)
5x
D
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Lines Intersecting Inside or Outside a Circle

• If two lines intersect a circle, there are three
  (3) places where the lines can intersect.

on the circle
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Inside the circle
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Outside the circle
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Lines Intersecting

• You know how to find angle and arc measures when
  lines intersect
• ON THE CIRCLE.
• You can use the following theorems to find the
  measures when the lines intersect
• INSIDE or OUTSIDE the circle.

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Theorem 10.13

• If two chords intersect in the interior of a
  circle, then the measure of each angle is one
  half the sum of the measures of the arcs
  intercepted by the angle and its vertical angle.

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Theorem 10.14

• If a tangent and a secant, two tangents or two
  secants intercept in the EXTERIOR of a circle,
  then the measure of the angle formed is one half
  the difference of the measures of the intercepted
  arcs.

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Theorem 10.14

• If a tangent and a secant, two tangents or two
  secants intercept in the EXTERIOR of a circle,
  then the measure of the angle formed is one half
  the difference of the measures of the intercepted
  arcs.

m?2 ½ m( - m )
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Theorem 10.14

• If a tangent and a secant, two tangents or two
  secants intercept in the EXTERIOR of a circle,
  then the measure of the angle formed is one half
  the difference of the measures of the intercepted
  arcs.

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m?3 ½ m( - m )
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Ex. 3 Finding the Measure of an Angle Formed by
Two Chords
106

• Find the value of x
• Solution
• x ½ (m m
• x ½ (106 174)
• x 140

x
174
Apply Theorem 10.13
Substitute values
Simplify
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Ex. 4 Using Theorem 10.14
200

• Find the value of x
• Solution
• 72 ½ (200 - x)
• 144 200 - x
• - 56 -x
• 56 x

x
72
Apply Theorem 10.14
Substitute values.
Multiply each side by 2.
Subtract 200 from both sides.
Divide by -1 to eliminate negatives.
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Ex. 4 Using Theorem 10.14
Because and make a whole
circle, m 360-92268
x
92

• Find the value of x
• Solution
• ½ (268 - 92)
• ½ (176)
• 88

Apply Theorem 10.14
Substitute values.
Subtract
Multiply
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Ex. 5 Describing the View from Mount Rainier

• You are on top of Mount Rainier on a clear day.
  You are about 2.73 miles above sea level. Find
  the measure of the arc that represents the
  part of Earth you can see.

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Ex. 5 Describing the View from Mount Rainier

• You are on top of Mount Rainier on a clear day.
  You are about 2.73 miles above sea level. Find
  the measure of the arc that represents the
  part of Earth you can see.

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Ex. 5 Describing the View from Mount Rainier

• and are tangent to the Earth. You can
  solve right ?BCA to see that m?CBA ? 87.9. So,
  m?CBD ? 175.8. Let m x using Trig
  Ratios

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175.8 ? ½(360 x) x 175.8 ? ½(360 2x)
175.8 ? 180 x x ? 4.2
Apply Theorem 10.14.
Simplify.
Distributive Property.
Solve for x.
?From the peak, you can see an arc about 4.
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Reminder

• Quiz after section 10.5
• Deficiencies go out the week of April 25-28 so
  all work is due no later than the end of this
  week.