Inscribed AnglesInscribed Angles
Challenge ProblemChallenge Problem
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ObjectivesObjectives
•Use inscribed angles to solve problems.
•Use properties of inscribed polygons.
Using Inscribed AnglesUsing Inscribed Angles
•An inscribed angle is anangle whose vertex is ona circle and whose sidescontain chords of thecircle. The arc that lies inthe interior of an inscribedangle and has endpointson the angle is called theintercepted arc of theangle.
Measure of an Inscribed AngleMeasure of an Inscribed Angle
Finding Measures of Arcs and Inscribed AnglesFinding Measures of Arcs and Inscribed Angles
•Find the measureof the blue arc orangle.
m = 2mQRS =
2(90°) = 180°
Finding Measures of Arcs and Inscribed AnglesFinding Measures of Arcs and Inscribed Angles
•Find the measureof the blue arc orangle.
m = 2mZYX =
2(115°) = 230°
Ex. 1: Finding Measures of Arcs and Inscribed AnglesEx. 1: Finding Measures of Arcs and Inscribed Angles
•Find the measureof the blue arc orangle.
m = ½ m
½ (100°) = 50°
100°
Ex. 2: Comparing Measures of Inscribed AnglesEx. 2: Comparing Measures of Inscribed Angles
•Find mACB,mADB, andmAEB.
The measure of eachangle is half themeasure of
m = 60°, so themeasure of eachangle is 30°
TheoremTheorem
•If two inscribed anglesof a circle interceptthe same arc, then theangles are congruent.
•C D
Ex. 3: Finding the Measure of an AngleEx. 3: Finding the Measure of an Angle
•It is given thatmE = 75°. Whatis mF?
•E and F bothintercept , soE F. So,mF = mE = 75°
75°
Using Properties of Inscribed PolygonsUsing Properties of Inscribed Polygons
•If all of the vertices ofa polygon lie on acircle, the polygon isinscribed in the circleand the circle iscircumscribed aboutthe polygon. Thepolygon is aninscribed polygonand the circle is acircumscribed circle.
TheoremTheorem
•If a right triangle is inscribed in a circle,then the hypotenuse is a diameter ofthe circle. Conversely, if one side ofan inscribed triangle is a diameter ofthe circle, then the triangle is a righttriangle and the angle opposite thediameter is the right angle.
•B is a right angle if and only if AC isa diameter of the circle.
TheoremTheorem
Ex. 5: Using TheoremsEx. 5: Using Theorems
2x°
Ex. 5: Using TheoremsEx. 5: Using Theorems
120°
80°
y°
z°
Ex. 5: Using TheoremsEx. 5: Using Theorems
120°
80°
y°
z°
Ex. 6: Using an Inscribed QuadrilateralEx. 6: Using an Inscribed Quadrilateral
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Ex. 6: Using an Inscribed QuadrilateralEx. 6: Using an Inscribed Quadrilateral
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Angle Measures and Segment Lengthsin CirclesAngle Measures and Segment Lengthsin Circles
Objectives:
1.To find the measures of s formed by chords,secants, & tangents.
2.To find the lengths of segments associated withcircles.
SecantsSecants
E
A
B
F
Theorem. The measure of an formed by 2 lines thatintersect inside a circle is
Measure of intercepted arcs
Theorem. The measure of an formed by 2 linesthat intersect outside a circle is
Smaller Arc
Larger Arc
x°
y°
1
x°
y°
1
2 Secants:
x°
y°
1
Tangent & a Secant
2 Tangents
3 cases:
Ex.1 & 2:Ex.1 & 2:
94°
112°
68°
104°
92°
268°
Lengths of Secants, Tangents, & ChordsLengths of Secants, Tangents, & Chords
2 Chords
2 Secants
Tangent & Secant
Ex. 3 & 4Ex. 3 & 4
3
7
5
15
8
Ex.5: 2 SecantsEx.5: 2 Secants
14
20
16
Ex.6: A little bit of everything!Ex.6: A little bit of everything!
9
12
8
60°
175°
What have we learned??What have we learned??
•When dealing with angle measures formed byintersecting secants or tangents you either addor subtract the intercepted arcs depending onwhere the lines intersect.
•There are 3 formulas to solve for segmentslengths inside of circles, it depends on whichsegments you are dealing with: Secants,Chords, or Tangents.
Q
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100°
30°
25°
Challenge #1Challenge #1
Find the value of the variable.
6
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16
Challenge #2Challenge #2
Find the value of the variable.
16
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6.5
Challenge #3Challenge #3
Q
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1
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A
100°
30°
25°
Find the value of the variable.
6
7
8
8
16
Find the value of the variable to the nearest tenth.
16
20
14
7
3
6.5
