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10.4 Other Angle Relationships in Circles PowerPoint PPT Presentation


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10.4 Other Angle Relationships in Circles. Geometry Mrs. Spitz Spring 2005. 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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10.4 Other Angle Relationships in Circles

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10.4 Other Angle Relationships in Circles

Geometry

Mrs. Spitz

Spring 2005


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


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


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


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°


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°


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


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


Inside the circle


Outside the circle


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.


m1 = ½ m + m

m2 = ½ m + m

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.


m1 = ½ m( - m )

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.


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 )


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.

3

m3 = ½ m( - m )


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

174°

Apply Theorem 10.13

Substitute values

Simplify


mGHF = ½ m( - m )

Ex. 4: Using Theorem 10.14

200°

  • Find the value of x

    Solution:

    72° = ½ (200° - x°)

    144 = 200 - x°

    - 56 = -x

    56 = x

72°

Apply Theorem 10.14

Substitute values.

Multiply each side by 2.

Subtract 200 from both sides.

Divide by -1 to eliminate negatives.


mGHF = ½ m( - m )

Ex. 4: Using Theorem 10.14

Because and make a whole circle, m =360°-92°=268°

92°

  • Find the value of x

    Solution:

    = ½ (268 - 92)

    = ½ (176)

    = 88

Apply Theorem 10.14

Substitute values.

Subtract

Multiply


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.


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.


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


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°.


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.


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