Download Chapter 10, Section 3: Inscribed Angles

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 their
properties to solve problems.
2. I can apply properties of inscribed polygons
to solve problems.
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Using Inscribed Angles
inscribed
angle
intercepted
arc
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°
Ex. 2: Finding Measures of Arcs and Inscribed Angles
Find the measure of the blue arc or angle.
m∠ZYX = 115°
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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°
Theorem 10.9
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
∠C ≅ ∠D
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?
SHOW ME
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
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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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Theorem 10.10
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.
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°
Theorem 10.11
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°
Find the value of each variable.
z°
DEFG is inscribed in a circle, so
opposite angles are
supplementary.
m∠E + m∠G = 180°
y + 120 = 180
y = 60
120°
80°
y°
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Example: Inscribed Polygon
2y°
In the diagram, ABCD is inscribed in circle P.
Find the measure of each angle.
3y°
3x°
ABCD is inscribed in a circle, so opposite
angles are supplementary.
3x + 3y = 180
5x + 2y = 180
2x°
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.
Solution
5x + 2y = 180.
5x + 2 (60 ­ x) = 180
5x + 120 ­ 2x = 180
3x = 60
x = 20
y = 60 ­ 20 = 40
Solutions:
x = 20 and y = 40, so m∠A = 80°, m∠B = 60°, m∠C = 100°, and m∠D = 120°
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Arcs and Angles
Graphic Organizer
Graphic Organizer: Review
Using the figure to the right, name one example for each of the following: 1. major arc
2. inscribed angle 3. central angle
4. intercepted arc
5. semicircle
6. minor arc
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Section 10.1 - 10.3 Review
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