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```10­4
Use Inscribed Angles
and Polygons
1
Inscribed Angle ­ Vertex lies on the circle, the sides are the chords.
A
C
Example: <ABC is an inscribed angle of circle P.
P
B
What is <APC?
Theorem 10­7 Measure of an Inscribed Angle Theorem The measure of an inscribed angle is one half the measure of its intercepted arc.
Example: If <ABC = 350, then
A
C
mAC = ______
mABC = _______
P
B
What type of angle is <APC? ____________
What is the m<APC? ___________
http://www.ronblond.com/M11/INSC.INSC.APPLET/index.html
2
Theorem 10­8 If 2 inscribed angles of a circle intercept the same arc, then the angles are congruent.
A
D
C
B
Examples:
1. In a regular inscribed 5 point star. Find the measure of each inscribed angle.
3
2.) A
B
m<ACB = 300
mAB = ________
C
D
4
Theorem 10­9
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.
Conversely, if 1 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.
A
m<ABCD = 900 iff AC is a diameter of the circle.
D
B
C
Theorem 10­12
A quadrilateral can be inscribed in a circle iff its opposite angles are supplementary.
Can you inscribe a 1. Square? 1. yes
2. Parallelogram?
2. No
3. Rectangle?
3. Yes
4. Trapezoid?
5. Isosceles Trapezoid?
6. Kite?
4. No
5. Yes
6. No
The above is
a
circumscribed circle