SECTION 10.4Inscribed Angles

An inscribed angle has a vertex on a circle and sides that contain chords of the circle.

In , ⨀C, QRS is an inscribed angle.

An intercepted arc has endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle. In ⨀C, minor arc is intercepted by QRS.QS

There are three ways that an angle can be inscribed in a circle.

For each of these cases, the following theorem holds true.

Example 1:

a) Find mX.

b) Find .mYX

1

21

862

43

m X mZW

2

2 52

104

mYX m Z

Example 1:

c) Find mC.

d) Find .mBC

1

21

942

47

m C mAD

2

2 48

96

mBC m A

Example 2:

a) Find mR.

R S R and S both intercept . mR = mS Definition of congruent angles

12x – 13 = 9x + 2 Substitutionx = 5 Simplify.

Answer: So, mR = 12(5) – 13 or 47º.

TQ

Example 2:

b) Find mI.

I J R and S both intercept . mI = mJ Definition of congruent angles

8x + 9 = 10x – 1 Substitutionx = 5 Simplify.

Answer: So, mI = 8(5) + 9 or 49º.

GH

Example 3: Write a two-column proof.

Given:Prove: ∆MNP ∆LOP

LO MN

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

LO MN Given

If minor arcs are congruent, then corresponding chords are congruent.

Definition of intercepted arc

Inscribed angles of the same arc are congruent.

LO MN

intercepts and

intercepts .

M NO

L NO

M L

Example 3: Write a two-column proof.

Given:Prove: ∆MNP ∆LOP

LO MN

Statements Reasons

5. 5.

6. 6.

Vertical angles are congruent.

AAS Congruence Theorem

MPN OPL

∆MNP ∆LOP

Example 4:

a) Find mB.

ΔABC is a right triangle because C inscribes a semicircle.

mA + mB + mC = 180 Angle Sum Theorem(x + 4) + (8x – 4) + 90 = 180 Substitution

9x + 90 = 180 Simplify.9x = 90 Subtract 90 from each

side.x = 10 Divide each side by 9.

Answer: So, mB = 8(10) – 4 or 76º.

Example 4:

b) Find mD.

ΔDEF is a right triangle because F inscribes a semicircle.

mD + mE + mF = 180 Angle Sum Theorem(2x + 6) + (8x + 4) + 90 = 180 Substitution

10x + 100 = 180 Simplify.10x = 80 Subtract 90 from each

side.x = 8 Divide each side by 10.

Answer: So, mD = 2(8) + 6 or 22º.

Example 5:

a) An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mS and mT.

Since TSUV is inscribed in a circle, opposite angles are supplementary.

mS + mV = 180 mU + mT = 180 mS + 90 = 180 (14x) + (8x + 4) = 180

mS = 90 22x + 4 = 18022x = 176

x = 8Answer: So, mS = 90º and mT = 8(8) + 4 or 68º.

Example 5:

b) An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mN.

Since LMNO is inscribed in a circle, opposite angles are supplementary.

mL + mN = 180 (11x) + (3x + 12) = 180

14x + 12 = 18014x = 168

x = 12Answer: So, mN = 3(12) + 12 or 48º.