Over Lesson 10–3
You found measures of interior angles of
polygons.
• Find measures of inscribed angles.
• Find measures of angles of inscribed
polygons.
• inscribed angle
• intercepted arc
Use Inscribed Angles to Find Measures
A. Find mX.
Answer: mX = 43
Use Inscribed Angles to Find Measures
B.
= 2(52) or 104
A. Find mC.
A. 47
B. 54
C. 94
D. 188
B.
A. 47
B. 64
C. 94
D. 96
Use Inscribed Angles to Find Measures
ALGEBRA Find mR.
R  S
R and S both intercept
.
mR  mS
Definition of congruent angles
12x – 13 = 9x + 2
Substitution
x =5
Simplify.
Answer: So, mR = 12(5) – 13 or 47.
ALGEBRA Find mI.
A. 4
B. 25
C. 41
D. 49
Use Inscribed Angles in Proofs
Write a two-column proof.
Given:
Prove: ΔMNP  ΔLOP
Proof:
Statements
LO  MN
Reasons
1. Given
2. If minor arcs are congruent,
then corresponding chords
are congruent.
Use Inscribed Angles in Proofs
Proof:
Statements
M intercepts
L intercepts
M  L
MPN  OPL
ΔMNP  ΔLOP
Reasons
and
.
3. Definition of intercepted arc
4. Inscribed angles of the
same arc are congruent.
5. Vertical angles are
congruent.
6. AAS Congruence Theorem
Find Angle Measures in Inscribed Triangles
ALGEBRA Find mB.
ΔABC is a right triangle because
C inscribes a semicircle.
mA + mB + mC = 180
(x + 4) + (8x – 4) + 90 = 180
9x + 90 = 180
9x = 90
Angle Sum Theorem
Substitution
Simplify.
Subtract 90 from each
side.
x = 10
Divide each side by 9.
Answer: So, mB = 8(10) – 4 or 76.
ALGEBRA Find mD.
A. 8
B. 16
C. 22
D. 28
Find Angle Measures
INSIGNIAS An insignia is an emblem that
signifies rank, achievement, membership, and so
on. The insignia shown is a quadrilateral
inscribed in a circle. Find mS and mT.
Find Angle Measures
Since TSUV is inscribed in a circle, opposite angles are
supplementary.
mS + mV = 180
mU + mT = 180
mS + 90 = 180
(14x) + (8x + 4) = 180
mS = 90
22x + 4 = 180
22x = 176
x =8
Answer: So, mS = 90 and mT = 8(8) + 4 or 68.
INSIGNIAS An insignia is an emblem that signifies
rank, achievement, membership, and so on. The
insignia shown is a quadrilateral inscribed in a
circle. Find mN.
A. 48
B. 36
C. 32
D. 28