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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


Lines intersecting inside or outside a circle l.jpg
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




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°.


Reminder l.jpg
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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