12.3 Inscribed Angles An angle whose vertex is on the circle and whose sides are chords of the...

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.

12

m B AC

Three Cases to Consider

Using the Inscribed Angle Theorem• What are the values of a and b?

1

2m PQT mPT

160

2a

120a 1

2m PRS mPS

1(120 30)2

b

75b

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?

1 90m 2 38m

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

2mPMQ m PQS

1(212)2

m PQS 106 m PQS

180m PQS m PQR 106 180m PQR

74m PQR

More Practice!!!!!

• Homework – Textbook p. 784 # 6 – 18 ALL.