Five-Minute Check (over Lesson 10–3)
NGSSS
Then/Now
New Vocabulary
Theorem 10.6: Inscribed Angle Theorem
Proof: Inscribed Angle Theorem (Case 1)
Example 1: Use Inscribed Angles to Find Measures
Theorem 10.7
Example 2: Use Inscribed Angles to Find Measures
Example 3: Use Inscribed Angles in Proofs
Theorem 10.8
Example 4: Find Angle Measures in Inscribed Triangles
Theorem 10.9
Example 5: Real-World Example: Find Angle Measures
Over Lesson 10–3
A. 60
0%
B
D. 90
A
0%
A
B
C
0%
D
D
C. 80
C
B. 70
A.
B.
C.
0%
D.
Over Lesson 10–3
A. 40
0%
B
D. 55
A
0%
A
B
C
0%
D
D
C. 50
C
B. 45
A.
B.
C.
0%
D.
Over Lesson 10–3
A. 40
0%
B
D. 55
A
0%
A
B
C
0%
D
D
C. 50
C
B. 45
A.
B.
C.
0%
D.
Over Lesson 10–3
A. 40
0%
B
D. 22.5
A
0%
A
B
C
0%
D
D
C. 25
C
B. 30
A.
B.
C.
0%
D.
Over Lesson 10–3
A. 24.6
0%
B
D. 30.2
A
0%
A
B
C
0%
D
D
C. 28.4
C
B. 26.8
A.
B.
C.
0%
D.
Over Lesson 10–3
A.
0%
B
D.
A
0%
A
B
C
0%
D
D
C.
A.
B.
C.
0%
D.
C
B.
MA.912.G.6.1 Determine the center of a
given circle. Given three points not on a line,
construct the circle that passes through
them. Construct tangents to circles.
Circumscribe and inscribe circles about and
within triangles and regular polygons.
MA.912.G.6.4 Determine and use measures
of arcs and related angles.
Also addresses MA.912.G.6.3.
You found measures of interior angles of
polygons. (Lesson 6–1)
• 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(252) or 104
A. Find mC.
A. 47
B. 54
0%
B
A
0%
A
B
C
0%
D
D
D. 188
C
C. 94
A.
B.
C.
0%
D.
B.
A. 47
B. 64
0%
B
A
0%
A
B
C
0%
D
D
D. 96
C
C. 94
A.
B.
C.
0%
D.
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
0%
B
A
0%
A
B
C
0%
D
D
D. 49
C
C. 41
A.
B.
C.
0%
D.
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
Write a two-column proof.
Given:
Prove: ΔABE  ΔDCE
Select the appropriate reason that goes
in the blank to complete the proof below.
Proof:
Statements
Reasons
AB  DC
1. Given
2. If minor arcs are congruent,
then corresponding chords
are congruent.
Proof:
Statements
D intercepts
A intercepts
D  A
DEC  BEA
ΔDCE ΔABE
Reasons
and
.
3. Definition of intercepted arc
4. Inscribed angles of the
same arc are congruent.
5. Vertical angles are
congruent.
6. ____________________
A. SSS Congruence Theorem
B. AAS Congruence Theorem
0%
B
A
0%
A
B
C
0%
D
D
D. Definition of congruent arcs
A.
B.
C.
0%
D.
C
C. Definition of congruent
triangles
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
D. 28
0%
B
A
0%
0%
C
C. 22
A
B
C
D
0%
D
A.
B.
C.
D.
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.
S + V = 180
S + V = 180
S + 90 = 180
(14x) + (8x + 4) = 180
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
D. 28
0%
B
A
0%
A
B
C
0%
D
D
C. 32
C
B. 36
A.
B.
C.
0%
D.