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Chapter 10, Section 3:Inscribed Angles

How can we apply properties of inscribed angles to help us choose a seat for the Hunger Games movie or Snow White with Julia Roberts?

Section 10.3 Goals

G-C.2 1. I can define inscribed angles and apply theirproperties to solve problems.

2. I can apply properties of inscribed polygons to solve problems.

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inscribed angle

intercepted arc

Using Inscribed Angles

inscribed angle - an angle whose vertex is on a circle and whose sides contain chords of the circle.

intercepted arc - the arc that lies in the interior of an inscribed angle and has endpoints on the angle.

Theorem 10.8:  Measure of an Inscribed Angle

If an angle is inscribed in a circle, then its measure is one half the measure of its intercepted arc.

m∠ADB = ½m

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Finding Measures of Arcs and Inscribed Angles Using Theorem 10.8

Find the measure of the blue arc or angle.

m         = 2m∠QRS =2(90°) = 180°

Find the measure of the blue arc or angle.

m∠ZYX = 115°

Ex. 2:  Finding Measures of Arcs and Inscribed Angles

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Comparing Measures of Inscribed Angles

Find m∠ACB, m∠ADB, and m∠AEB.

Using Theorem 10.8 we can say that

the measure of each angle is half the measure of      .  If m       = 60°, the measure of each angle is 30°

If two inscribed angles of a circle intercept the same arc, then the angles are congruent.∠C ≅ ∠D

Theorem 10.9

m∠C = 75°

Both ∠C and ∠D intercept  So...

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Real Life Example of Inscribed Angles

When you go to the movies, you want to be close to the movie screen, but you don’t want to have to move your eyes too much to see the edges of the picture.  

If E and G are the ends of the screen and you are at F, m∠EFG is called your viewing angle.

You decide that the middle of the sixth row has the best viewing angle.  If someone else is sitting there, where else can you sit to have the same viewing angle?

Solution:  Draw the circle that is determined by the endpoints of the screen and the sixth row center seat.  Any other location on the circle will have the same viewing angle.  Theorem 10.9

SHOW ME

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Homework Textbook Page 617, #10 - 20 even

Inscribed Polygons

If all the vertices of a polygon lie on a circle, the polygon is said to be inscribed and the circle is said to be circumscribed.

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If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

∠B is a right angle if and only if AC is a diameter of the circle.

Theorem 10.10

Find the value of each variable.

AB is a diameter.  So, ∠C is a right angle and m∠C = 90°2x° = 90°x = 45

2x°

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.D, E, F, and G lie on some circle, C, if and only if m∠D + m∠F = 180° and m∠E + m∠G = 180°

Theorem 10.11

z° 120°

y°

80°

Find the value of each variable.

DEFG is inscribed in a circle, so opposite angles are supplementary.m∠E + m∠G = 180°y + 120 = 180y = 60

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2y°

3y°

2x°

3x°

In the diagram, ABCD is inscribed in circle P. Find the measure of each angle.

ABCD is inscribed in a circle, so opposite angles are supplementary.3x + 3y = 1805x + 2y = 180

To solve this system of linear equations, you can solve the first equation for y to get y = 60 ­ x.  Substitute this expression into the second equation.

Example: Inscribed Polygon

5x + 2y = 180.5x + 2 (60 ­ x) = 1805x + 120 ­ 2x = 1803x = 60x = 20y = 60 ­ 20 = 40

x = 20 and y = 40, so m∠A = 80°, m∠B = 60°, m∠C = 100°, and m∠D = 120°

Solutions:

Solution

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Arcs and AnglesGraphic Organizer

Using the figure to the right, name one example for each of the following:  

1.      major arc2.     inscribed angle 3.     central angle4.     intercepted arc5.     semicircle6.     minor arc

Graphic Organizer: Review

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Section 10.1 - 10.3 Review