Chapter 10 – Circles Section 10.3 – Inscribed Angles

Unit Goal Use inscribed angles to solve problems.

D A B C

Basic Definitions INSCRIBED ANGLE – an angle whose vertex is on the circle INTERCEPTED ARC – the arc whose endpoints are are on the inscribed angle D

DCB

is an inscribed angle.

DB

is the intercepted arc.

A B C

What Is the Measure of an Inscribed Circle?

D C B A E

Theorem 10.8

Measure of an Inscribed Angle The measure of an inscribed angle is ½ of its intercepted arc.

B

m BDC

 1 2

mBC

A C D

Example I Find the measure of the angle or arc: C D A A B

m BDC

 140 

mBIC

 C 20º B

m CDB

 D

D Example Find the measure of the angle or arc: B A C

m BDC



mBEC

 E A D 50º C

mBC

 B

D J K Example B 60º C

m BDC



m BJC m BKC

 

Theorem 10.9

If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

B 60º C D J K

Properties of Inscribed Polygons If all the vertices of a polygon lie on a circle, the polygon is INSCRIBED in the circles and the circle is CIRCUMSCRIBED about the polygon

Theorems About Inscribed Polygons 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

Theorem 10.11

A quadrilateral can be inscribed in a circle iff its opposite angles are supplementary D, E, F, and G lie on some circle C iff m

In the diagram, ABCD is inscribed in circle P. Find the measure of each angle.

ABCD is inscribed in a circle, so opposite angles are supplementary 3x + 3y = 180 and 5x+ 2y = 180 3x + 3y = 180 (solve for x) - 3y -3y 3x = -3y + 180 3 3 x = -y + 60 Substitute Example D A 2y 3y equation 5x + 2y = 180 5 (-y + 60) + 2y = 180 -5y + 300 + 2y = 180 -3y = -120 y = 40 x = -y + 60 x = -40 + 60 = 20 P 5x 3x B

Example (cont.) x = 20, y = 40 m

HW Assignment p. 616-617 (4 – 28 even)