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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 l.jpg

10.4 Other Angle Relationships in Circles

Geometry

Mrs. Spitz

Spring 2005


Objectives assignment l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 measures6 l.jpg

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 l.jpg

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


Inside the circle l.jpg

Inside the circle


Outside the circle l.jpg

Outside the circle


Lines intersecting l.jpg

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.


Theorem 10 13 l.jpg

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.


Theorem 10 14 l.jpg

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 1414 l.jpg

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 1415 l.jpg

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 l.jpg

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


Ex 4 using theorem 10 14 l.jpg

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.


Ex 4 using theorem 10 1418 l.jpg

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 l.jpg

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 rainier20 l.jpg

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 rainier21 l.jpg

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


Slide22 l.jpg

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.


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