This document covers theorems and examples relating to arcs and chords in circles. It states four theorems: 1) Congruent minor arcs have congruent chords, 2) A diameter or radius perpendicular to a chord bisects the chord and arc, 3) The perpendicular bisector of a chord is a diameter or radius, and 4) Congruent chords are equidistant from the center. It then provides five examples applying these theorems to find measures of arcs, chords, and radii given various circle properties. The document concludes with sample problems for students to practice these concepts.

1. Section 10-3
Arcs and Chords
Monday, May 14, 2012

2. Essential Questions
• How do you recognize and use
relationships between arcs and chords?
• How do you recognize and use
relationships between arcs, chords, and
diameters?
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3. Theorems
10.2 - Congruent Minor Arcs:
10.3 - Perpendicularity:
10.4 - Perpendicularity:
10.5 - Congruent Chords:
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4. Theorems
10.2 - Congruent Minor Arcs: In the same or congruent
circles, two minor arcs are congruent IFF their
corresponding chords are congruent
10.3 - Perpendicularity:
10.4 - Perpendicularity:
10.5 - Congruent Chords:
Monday, May 14, 2012

5. Theorems
10.2 - Congruent Minor Arcs: In the same or congruent
circles, two minor arcs are congruent IFF their
corresponding chords are congruent
10.3 - Perpendicularity: If a diameter or radius of a circle is
perpendicular to a chord, then it bisects the chord and its
arc
10.4 - Perpendicularity:
10.5 - Congruent Chords:
Monday, May 14, 2012

6. Theorems
10.2 - Congruent Minor Arcs: In the same or congruent
circles, two minor arcs are congruent IFF their
corresponding chords are congruent
10.3 - Perpendicularity: If a diameter or radius of a circle is
perpendicular to a chord, then it bisects the chord and its
arc
10.4 - Perpendicularity: The perpendicular bisector of a
chord is a diameter or radius of the circle
10.5 - Congruent Chords:
Monday, May 14, 2012

7. Theorems
10.2 - Congruent Minor Arcs: In the same or congruent
circles, two minor arcs are congruent IFF their
corresponding chords are congruent
10.3 - Perpendicularity: If a diameter or radius of a circle is
perpendicular to a chord, then it bisects the chord and its
arc
10.4 - Perpendicularity: The perpendicular bisector of a
chord is a diameter or radius of the circle
10.5 - Congruent Chords: In the same or congruent circles,
two chords are congruent IFF they are equidistant from
the center
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8. Example 1
 = 90°. Find mAB.
In  X, AB ≅ CD and mCD 
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9. Example 1
 = 90°. Find mAB.
In  X, AB ≅ CD and mCD 
 = 90°
mAB
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10. Example 2
 ≅ YZ . Find WX.
In the figure,  A ≅ B and WX 
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11. Example 2
 ≅ YZ . Find WX.
In the figure,  A ≅ B and WX 
7x − 2 = 5x + 6
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12. Example 2
 ≅ YZ . Find WX.
In the figure,  A ≅ B and WX 
7x − 2 = 5x + 6
2x = 8
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13. Example 2
 ≅ YZ . Find WX.
In the figure,  A ≅ B and WX 
7x − 2 = 5x + 6
2x = 8
x=4
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14. Example 2
 ≅ YZ . Find WX.
In the figure,  A ≅ B and WX 
7x − 2 = 5x + 6
2x = 8
x=4
WX = 7(4) − 2
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15. Example 2
 ≅ YZ . Find WX.
In the figure,  A ≅ B and WX 
7x − 2 = 5x + 6
2x = 8
x=4
WX = 7(4) − 2
WX = 28 − 2
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16. Example 2
 ≅ YZ . Find WX.
In the figure,  A ≅ B and WX 
7x − 2 = 5x + 6
2x = 8
x=4
WX = 7(4) − 2
WX = 28 − 2
WX = 26
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17. Example 3
 =150°. Find mDE.
In  G, mDEF 
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18. Example 3
 =150°. Find mDE.
In  G, mDEF 
 = 1 mDEF
mDE 
2
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19. Example 3
 =150°. Find mDE.
In  G, mDEF 
 = 1 mDEF
mDE 
2
 = 1 (150)
mDE
2
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20. Example 3
 =150°. Find mDE.
In  G, mDEF 
 = 1 mDEF
mDE 
2
 = 1 (150)
mDE
2
 = 75°
mDE
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21. Example 4
In  C, AB =18 inches and EF = 8 inches. Find CD.
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22. Example 4
In  C, AB =18 inches and EF = 8 inches. Find CD.
CF is a radius.
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23. Example 4
In  C, AB =18 inches and EF = 8 inches. Find CD.
CF is a radius.
a +b =c
2 2 2
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24. Example 4
In  C, AB =18 inches and EF = 8 inches. Find CD.
CF is a radius.
a +b =c
2 2 2
4 +b =9
2 2 2
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25. Example 4
In  C, AB =18 inches and EF = 8 inches. Find CD.
CF is a radius.
a +b =c
2 2 2
4 +b =9
2 2 2
16 + b = 81
2
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26. Example 4
In  C, AB =18 inches and EF = 8 inches. Find CD.
CF is a radius.
a +b =c
2 2 2
4 +b =9
2 2 2
16 + b = 81
2
b = 65
2
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27. Example 4
In  C, AB =18 inches and EF = 8 inches. Find CD.
CF is a radius.
a +b =c
2 2 2
4 +b =9
2 2 2
16 + b = 81
2
b = 65
2
b = 65
Monday, May 14, 2012

28. Example 4
In  C, AB =18 inches and EF = 8 inches. Find CD.
CF is a radius.
a +b =c
2 2 2
4 +b =9
2 2 2
16 + b = 81
2
b = 65
2
b = 65 inches or ≈ 8.06 inches
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29. Example 5
In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ.
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30. Example 5
In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ.
4x − 3 = 2x + 3
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31. Example 5
In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ.
4x − 3 = 2x + 3
2x = 6
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32. Example 5
In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ.
4x − 3 = 2x + 3
2x = 6
x=3
Monday, May 14, 2012

33. Example 5
In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ.
4x − 3 = 2x + 3
2x = 6
x=3
PQ = 4(3) − 3
Monday, May 14, 2012

34. Example 5
In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ.
4x − 3 = 2x + 3
2x = 6
x=3
PQ = 4(3) − 3
PQ = 12 − 3
Monday, May 14, 2012

35. Example 5
In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ.
4x − 3 = 2x + 3
2x = 6
x=3
PQ = 4(3) − 3
PQ = 12 − 3
PQ = 9
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36. Check Your
Understanding
p. 704 #1 - 6
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37. Problem Set
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38. Problem Set
p. 705 #7-33 odd, 45, 49, 51
"I may not have gone where I intended to go, but I think I
have ended up where I needed to be." - Douglas Adams
Monday, May 14, 2012