Warm-Up:1) simplify: (x – 1)²

2) Factor: 10r² – 35r

3) Factor: t² + 12t + 36

4) Solve: 2x² – 3x + 1 = x² + 2x – 3

5) Find the radius of a circle with a 30 cm chord 20 cm away from the center of the circle.

11.3: Inscribed Angles

Inscribed AngleAngle with its vertex on a circle and sides are

chords of the circle

DEF is inscribed in G

DEF intercepts DF

• DF is intercepted by DEF

Inscribed Angle Theorem:

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

(or arc measure is twice

inscribed angle measure)

• If DF=76°,

then mDEF=38°

Corollary 1:Two inscribed angles that intercept the same

arc are congruent.

1 & 6 intercept WX,

so 1 6

• If m1=47°,

then WX=94°,

and m6=47°

Corollary 2:An angle inscribed in a semicircle is a right

angle.

• BD is a diameter

• BCD is a semicircle,

so BAD = 90°

• BAD is a semicircle,

so BCD = 90°

Corollary 3:If a quadrilateral is inscribed in a circle, then

its opposite angles are supplementary.

• Quadrilateral EFGH is

inscribed in N

E + G = 180F + H = 180

• If mE=125°,

• If mF=87°,

then mG=55°then mH=93°

Theorem 11-10:The measure of an angle formed by a tangent and a chord is one-half the measure of the intercepted arc.

W

Z

XY

V

115°

245°

11.4: Angle Measures

SecantA line that intersects a circle in exactly 2 points

Line AB is a secant to N

A

B

N

Theorem 11.11a:The measure of an angle formed by 2 lines intersecting inside of a circle is ½ the sum of the measures of its intercepted arcs.

A

BC

D

164° 22°

Theorem 11.11b:The measure of the angle formed by 2 lines intersecting outside of a circle is ½ the difference of the measures of the intercepted arcs.

i) 2 secants:

D

B

C

A

E

72°

15°x°

Theorem 11.11b:The measure of the angle formed by 2 lines intersecting outside of a circle is ½ the difference of the measures of the intercepted arcs.

ii) secant-tangent:

K

L

JM

146°

54°x°

Theorem 11.11b:The measure of the angle formed by 2 lines intersecting outside of a circle is ½ the difference of the measures of the intercepted arcs.

iii) 2 tangents:S

RT

U266°

x°