1. Arcs and Chords
Chapter 10-3

2. Recognize and use relationships between arcs and chords.
Recognize and use relationships between chords and diameters.
inscribed
circumscribed
Lesson 3 MI/Vocab

3. 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 3 CA

4. Chord Theorems
In the same circle or  circles, 2 minor arcs are   their corresponding chords are  B A C D E
BC  ED  BC  ED

5. PROOF Write a two-column proof.
Prove Theorem 10.2
PROOF Write a two-column proof.
Prove:
Given:
is a semicircle.
Lesson 3 Ex1

6. 3. In a circle, if 2 chords are , corr. minor arcs are . 3.
Prove Theorem 10.2
Answer:
Proof:
Statements Reasons 1.
1. Given
is a semicircle.
2. Def. of semicircle 2.
3. In a circle, if 2 chords are , corr. minor arcs are . 3.
4. Def. of arcs 4.
5. Def. of arc measure 5.
Lesson 3 Ex1

7. 6. Arc Addition Postulate
Prove Theorem 10.2
Answer:
Statements Reasons 6.
6. Arc Addition Postulate 7.
7. Substitution 8.
8. Subtraction Property and simplify 9.
9. Division Property
10.
10. Def. of arc measure
11.
11. Substitution
Lesson 3 Ex1

8. PROOF Choose the best reason to complete the following proof.
Given:
Prove:
Lesson 3 CYP1

9. 2. In a circle, 2 minor arcs are , chords are .
Proof:
Statements Reasons 1. 2. 3. 4.
1. Given
2. In a circle, 2 minor arcs are , chords are .
3. ______
4. In a circle, 2 chords are , minor arcs are .
Lesson 3 CYP1

10. A. Segment Addition Postulate B. Definition of 
C. Definition of Chord
D. Transitive Property A B C D
Lesson 3 CYP1

11. Inscribed Polygons 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

12. A regular hexagon is inscribed in a circle as part of a logo for an advertisement. If opposite vertices are connected by line segments, what is the measure of angle P in degrees?
Since connecting the opposite vertices of a regular hexagon divides the hexagon into six congruent triangles, each central angle will be congruent. The measure of each angle is 360 ÷ 6 or 60.
Answer: 60
Lesson 3 Ex2

13. ADVERTISING A logo for an advertising campaign is a pentagon that has five congruent central angles. Determine whether
A. yes
B. no
C. cannot be determined A B C
Lesson 3 CYP2

14. Chord Theorems
If the diameter of a circle is  to a chord,  the diameter bisects the chord and its arc A B C D
AD  DC
AB  BC

15. Radius Perpendicular to a Chord
Since radius is perpendicular to chord
Arc addition
Substitution
Substitution
Subtraction
Lesson 3 Ex3

16. Radius Perpendicular to a Chord
A radius perpendicular to a chord bisects it.
Def of seg bisector 10 8
Lesson 3 Ex3

17. Use the Pythagorean Theorem to find WJ.
JK = 8, WK = 10
Simplify.
Subtract 64 from each side.
Take the square root of each side. 8 10
Segment Addition Postulate
WJ = 6, WL = 10
Subtract 6 from each side. 6
Lesson 3 Ex3

18. A. 35
B. 70
C. 105
D. 145 A B C D
Lesson 3 CYP3

19. A. 15
B. 5
C. 10
D. 25 A B C D
Lesson 3 CYP3

20. Chord Theorems
In the same circle or  circles, 2 chords are   they are equidistant from the center.
EF  EG  AB  CD
& AB  CD F G E C D A B

21. Chords Equidistant from Center24
Pythagorean Theorem 15 12 9 24
Lesson 3 Ex4

22. A. 12
B. 36
C. 72
D. 32 A B C D
Lesson 3 CYP4

23. A. 12
B. 36
C. 72
D. 32 A B C D
Lesson 3 CYP4

24. Chord Theorems Sample Problem
AD  DC
Solve for x + y
3x + 7 = 5x + 3
4 = 2x
2=x
AD = 3x + 7; DC = 5x +3
m AB = 4y + 8; m AEC = 96 A C D E
AB  BC
AB  ½ AC
m AC = m AEC
m AC = 96
4y + 8 = ½ (96)
4y + 8 = 48
4y = 40
y = 10 B

25. Homework
Chapter 10.3
Pg – 31 all