10.4: inscribed angles
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Transcript 10.4: inscribed angles
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 mX.
Answer: mX = 43
Use Inscribed Angles to Find Measures
B.
= 2(52) or 104
A. Find mC.
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 mR.
R S
R and S both intercept
.
mR mS
Definition of congruent angles
12x – 13 = 9x + 2
Substitution
x =5
Simplify.
Answer: So, mR = 12(5) – 13 or 47.
ALGEBRA Find mI.
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 mB.
ΔABC is a right triangle because
C inscribes a semicircle.
mA + mB + mC = 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, mB = 8(10) – 4 or 76.
ALGEBRA Find mD.
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 mS and mT.
Find Angle Measures
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 = 180
22x = 176
x =8
Answer: So, mS = 90 and mT = 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 mN.
A. 48
B. 36
C. 32
D. 28