CHAPTER 10.4 & 10.5

10.4 USING INSCRIBED ANGLES AND POLYGONS Inscribed angle- an angle whose vertex is on

a circle and whose sides contain chords of the circle.

Intercepted arc- an arc that lies in the interior of an inscribed angle and has endpoints on the angle.

Intercepted arc

Inscribed angle

Measure of an inscribed angle theorem- The measure of an inscribed angle is one half the measure of its intercepted arc.

mADB = 12 mAB

B

A

CD

Example

Find a. mD =b. mAB =

35 50

A

CDE

B

EXAMPLE

Find a. mDEB =b. mDB =C. DAB

52A

DE

B

THEOREM If 2 inscribed angles of a circle intercept the

same arc, then the angles are congruent.

Inscribed polygon- a polygon in which all the vertices lie on the circle

Circumscribed circle- the circle that contains the vertices of the polygon

THEOREM A quadrilateral can be inscribed in a circle if

and only if its opposite angles are supplementary

mA + mC = 180mB + mD = 180

B

A

DC

EXAMPLE Find x and y.

yx

10060

B

A

DC

EXAMPLE Find x and y.

17y 7x

5x19y

BA

DC

THEOREM 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.

D

A

C

B

10.5 APPLYING OTHER ANGLE RELATIONSHIPS IN CIRCLES Theorem- If a tangent and a chord intersect at a point

on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

m2 =12 mACB

m1= 12 mAB

2 1A

BC

INTERSECTING LINES AND CIRCLES

outside the circleinside the circleon the circle

ANGLES IN THE CIRCLE THEOREM If the chords intersect inside a circle, then the

measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle

m2 = 12 mAD + mBC

m1 = 12 mDC + mAB

21

A

CB

D

ANGLES OUTSIDE THE CIRCLE THEOREM If a tangent and a secant, 2 tangents, or 2

secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

321