Transcript 12.3 Inscribed Angles - Cardinal O'Hara High School

12.3 Inscribed Angles
• An angle whose vertex is on the circle and whose
sides are chords of the circle is an inscribed angle.
• An arc with endpoints on the sides of an inscribed
angle, and its points in the interior of the angle is an
intercepted arc.
Theorem 12.11 Inscribed Angle Theorem
• The measure of an inscribed angle if half the
measure of its intercepted arc.
1
mB  AC
2
Three Cases to Consider
Using the Inscribed Angle Theorem
• What are the values of a and b?
1
mPQT  mPT
2
1
60  a
2
a  120
1
mPRS  mPS
2
1
b  (120  30)
2
b  75
Corollaries to Theorem 12.11: The Inscribed Angle
Theorem
• Corollary 1 – two inscribed angles that intercept the
same arc are congruent.
• Corollary 2 – an angle inscribed in a semicircle is a
right angle.
• Corollary 3 – the opposite angles of a quadrilateral
inscribed in a circle are supplementary.
Using Corollaries to Find Angle Measures
• What is the measure of each numbered angle?
m1  90
m2  38
Theorem 12.12
• The measure of an angle formed by a
tangent and a chord is half the measure of
the intercepted arc.
Using Arc Measure
• In the diagram, line SR is tangent to the circle at Q. If
the measure of arc PMQ is 212, what is the measure
of angle PQR?
1
mPMQ  mPQS
2
1
(212)  mPQS
2
106  mPQS
mPQS  mPQR  180 mPQR  74
106  mPQR  180
More Practice!!!!!
• Homework – Textbook p. 784 # 6 – 18
ALL.