1. Five-Minute Check (over Lesson 10–3) 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
Lesson Menu

2. A. 60
B. 70
C. 80
D. 90
5-Minute Check 1

3. A. 40
B. 45
C. 50
D. 55
5-Minute Check 2

4. A. 40
B. 45
C. 50
D. 55
5-Minute Check 3

5. A. 40
B. 30
C. 25
D. 22.5
5-Minute Check 4

6. A. 24.6
B. 26.8
C. 28.4
D. 30.2
5-Minute Check 5

7. A. B. C. D.
5-Minute Check 6

8. Splash Screen

9. 10-4 Inscribed Angles (Pg. 709)
TARGETS
Find measures of inscribed angles.
Find measures of angles of inscribed polygons.
Then/Now

10. Content Standards Mathematical Practices 4 Model with mathematics.
G-C.2 Identify and describe relationships among inscribed angles, radii, and chords.
G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Mathematical Practices
4 Model with mathematics.
1 Make sense of problems and persevere in solving them.
Then/Now

11. You found measures of interior angles of polygons. (Lesson 6–1)
Find measures of inscribed angles.
Find measures of angles of inscribed polygons.
Then/Now

12. inscribed angle
intercepted arc
Vocabulary

13. Concept

14. Concept

15. A. Find mX. Answer: mX = 43 Use Inscribed Angles to Find Measures
Example 1

16. Use Inscribed Angles to Find Measures
= 2(52) or 104
Example 1

17. A. Find mC.
A. 47
B. 54
C. 94
D. 188
Example 1

18. B.
A. 47
B. 64
C. 94
D. 96
Example 1

19. Concept

20. R  S R and S both intercept .
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.
Example 2

21. ALGEBRA Find mI.
A. 4
B. 25
C. 41
D. 49
Example 2

22. Write a two-column proof. Given: Prove: ΔMNP  ΔLOP
Use Inscribed Angles in Proofs
Write a two-column proof.
Given:
Prove: ΔMNP  ΔLOP
Proof:
Statements Reasons
1. Given
LO  MN 2. If minor arcs are congruent, then corresponding chords are congruent.
Example 3

23. 3. Definition of intercepted arc M intercepts and L intercepts .
Use Inscribed Angles in Proofs
Proof:
Statements Reasons
3. Definition of intercepted arc
M intercepts and L intercepts
M  L Inscribed angles of the same arc are congruent.
MPN  OPL 5. Vertical angles are congruent.
ΔMNP  ΔLOP 6. AAS Congruence Theorem
Example 3

24. 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
1. Given
AB  DC 2. If minor arcs are congruent, then corresponding chords are congruent.
Example 3

25. 3. Definition of intercepted arc D intercepts and A intercepts .
Proof:
Statements Reasons
3. Definition of intercepted arc
D intercepts and A intercepts
D  A 4. Inscribed angles of the same arc are congruent.
DEC  BEA 5. Vertical angles are congruent.
ΔDCE  ΔABE 6. ____________________
Example 3

26. A. SSS Congruence Theorem B. AAS Congruence Theorem
C. Definition of congruent triangles
D. Definition of congruent arcs
Example 3

27. Concept

28. ΔABC is a right triangle because C inscribes a semicircle.
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 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, mB = 8(10) – 4 or 76.
Example 4

29. ALGEBRA Find mD.
A. 8
B. 16
C. 22
D. 28
Example 4

30. Concept

31. 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.
Example 5

32. Answer: So, mS = 90 and mT = 8(8) + 4 or 68.
Find Angle Measures
Since TSUV is inscribed in a circle, opposite angles are supplementary.
mS + mV = mU + mT = 180
mS = 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.
Example 5

33. 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
Example 5

34. End of the Lesson