1. Inscribed Angles
Chapter 10-4

2. Find measures of inscribed angles.
Find measures of angles of inscribed polygons.
intercepted
Standard Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. (Key)
Standard Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. (Key)
Lesson 4 MI/Vocab

3. Inscribed Angles
Def: An angle whose vertex is on a circle and whose sides contain chords
The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the Intercepted Arc of the angle

4. Inscribed Angles X = ½ Y Y = 2X
The measure of an inscribed angle is one half of its intercepted arc. X Y
X = ½ Y
Y = 2X

5. Measures of Inscribed Angles20 40
a a = 360
2a = 360
2a = 192
a = 96 a
108
Lesson 4 Ex1

6. Measures of Inscribed Angles20 40
m1 =
m2 =
m3 =
m4 =
m5 = 96 96
180 – ( ) = 106
108
Lesson 4 Ex1

7. A. 30
B. 60
C. 15
D. 120 A B C D
Lesson 4 CYP1

8. A. 110
B. 55
C. 125
D. 27.5 A B C D
Lesson 4 CYP1

9. A. 30
B. 80
C. 40
D. 10 A B C D
Lesson 4 CYP1

10. A. 110
B. 55
C. 125
D. 27.5 A B C D
Lesson 4 CYP1

11. A. 110
B. 55
C. 125
D. 27.5 A B C D
Lesson 4 CYP1

12. Inscribed Angles mX = m Y
If two inscribed angles of a circle intercept the same arc (or  arcs), then the angles are . X Y
mX = m Y

13. Proof with Inscribed Angles
Given:
Prove: ΔPJK  ΔEHG

14. Proof with Inscribed Angles
Statements Reasons
1. Given 1. 2.
2. If 2 chords are , corr. minor arcs are . 3.
3. Definition of intercepted arc 4.
4. Inscribed angles of arcs are . 5.
5. Right angles are congruent.
6. ΔPJK  ΔEHG
6. AAS
Lesson 4 Ex2

15. Choose the best reason to complete the following proof.
Given:
Prove: ΔCEM  ΔHJM
Lesson 4 CYP2

16. 3. Vertical angles are congruent. 4. Radii of a circle are congruent.
1. Given
2. ______
3. Vertical angles are congruent.
4. Radii of a circle are congruent.
5. ASA
Proof:
Statements Reasons 1. 2. 3. 4.
5. ΔCEM  ΔHJM
Alternate Interior Angle Theorem
Substitution
C. Definition of  angles
D. Inscribed angles of  arcs are .
Lesson 4 CYP2

17. 9th and 10th grade teachers should cover this topic briefly.
Inscribed Arcs and Probability
This Topic will not be on the chapter 10 test. But Geometric probability will appear on the CST.
9th and 10th grade teachers should cover this topic briefly.
Lesson 4 Ex3

18. 9th and 10th grade teachers should cover this topic briefly.
Inscribed Arcs and Probability
This Topic will not be on the chapter 10 test. But Geometric probability will appear on the CST.
9th and 10th grade teachers should cover this topic briefly.
The probability that is the same as the probability of L being contained in
Lesson 4 Ex3

19. 9th and 10th grade teachers should cover this topic briefly.
This Topic will not be on the chapter 10 test. But Geometric probability will appear on the CST.
9th and 10th grade teachers should cover this topic briefly.
A. B.
C. D. A B C D
Lesson 4 CYP3

20. Inscribed Angles A 90o inscribed angle intercepts a 180o arc
C is a right   AB is a diameter of the circle C B A
R = 2.5 5 3 4

21. Lesson 4 TH3

22. Lesson 4 Ex4

23. A. 45
B. 90
C. 180
D. 80 A B C D
Lesson 4 CYP4

24. A. 17
B. 76
C. 60
D. 42 A B C D
Lesson 4 CYP4

25. A. 17
B. 76
C. 60
D. 42 A B C D
Lesson 4 CYP4

26. A. 73
B. 30
C. 60
D. 48 A B C D
Lesson 4 CYP4

27. Inscribed Polygons
Reminder: If all the vertices of a polygon lie on the circle
The polygon is inscribed in the circle
The circle is circumscribed about the polygon
A quadrilateral can be inscribed in a circle  its opposite angles are supplementary
Supplementary—adds up to 180o

28. 3x + 2 2x-7 Inscribed Polygons (3x + 2) + (2x –7) =180 5x – 5 = 180

29. Angles of an Inscribed Quadrilateral
Draw a sketch of this situation.
Lesson 4 Ex5

30. Angles of an Inscribed Quadrilateral
To find we need to know
To find first find
Inscribed Angle Theorem
Sum of arcs in circle = 360
Subtract 174 from each side.
Lesson 4 Ex5

31. Angles of an Inscribed Quadrilateral
Inscribed Angle Theorem
Substitution
Divide each side by 2.
Since we now know three angles of a quadrilateral, we can easily find the fourth.
mQ + mR + mS + mT = ° in a quadrilateral
mT = 360 Substitution
mT = 78 Subtraction
Answer: mS = 93; mT = 78
Lesson 4 Ex5

32. A. 99
B. 104
C. 81
D. 76 A B C D
Lesson 4 CYP5

33. A. 99
B. 104
C. 81
D. 76 A B C D
Lesson 4 CYP5

34. Homework
Chapter 10-4
Pg 583
6 – 8, 15 – 21, 24 – 31 all