1. 10.1 Exploring Angles in a Circle
Math 9

2. Chord
A line segment connecting any two points on a circle is called a chord. In the figure above, AB is a chord. The longest chord in a circle is the diameter. The radius is not a chord because it does not have 2 endpoints on the circle. B A

3. Arc
An arc is a portion of the circumference. A major arc is an arc which covers more than half the circumference; a minor arc covers less than half the circumference.
EF is a minor arc E F

4. Inscribed Angle
BCA is an inscribed angle. Its vertex is on the circle and its arms are chords. BCA is said to subtend arc AB (or chord AB) Subtends –the angel stands on or is formed by the end points of the arc (or chord) C
Vertex C
Chord BC
Chord AC B
Arc AB A

5. Central Angle
EFG is a central angle. Its vertex is at the center and its arms are radii.  EFG is said to subtend arc EG (or chord EG) Note: Every time you see a central angle, you should actually see two of them – one that subtends a minor arc and another one that subtends a major arc. G F E

6. Use a protractor to measure each of the angles indicated in the circles below (Point E is the center): ABD = ABD = ACD = ACD = AED = AED = 66
132 29 58

7. Properties of Inscribed and Central Angles
They have the same angle
Properties of Inscribed and Central Angles
Inscribed angles subtending the same arc are congruent
The measure of the central angle is equal to twice the measure of the inscribed angle subtending the same arc. This could also be stated as the inscribed angle is half the central angle subtending the same arc

8. Indicate in the diagrams below which angles have the relationship that one is twice the other.x x 2x x 2x 2x 2x x

9. Inscribed Angle on a Diameter
What type of angle is  ABC ? a straight angle
How big is ABC ? 180
What is the relationship between  ABC and  ADC?
ABC = 2·  ADC
An inscribed angle subtending a diameter is 90. D C B A

10. Jigsaw

11. Determine the size of each of the angles (In each of the diagrams O is the center).
X is 110
X = 2(55)
because the central angle is equal to twice the measure of the inscribed angle subtending the same arc

12. Z subtends the same chord as the angle that is 30 therefore it is also 30. Y subtends the same chord as the angle that is 85 therefore it is also 85. The angles of a triangle add up to 180 therefore 180 = X X= 65

13. X is 200 X = 2(100) because the central angle is equal to twice the measure of the inscribed angle subtending the same arc Angles on a point add up to 360, therefore X + Y = 360 Y = 160

14. X = 50 Y = 100

15. Angle across from the diameter is 90. 180 = 90 + 28 + X X = 62
180/2 = 90
Angle across from the diameter is 90. 180 = X X = 62
180

16. This angle is congruent with X
= 50
180 = X 130 = 2X X = 65

17. X = 45

18. X + 20 = 90 X = 70 Y = 180 – (X + X +20) = 180 – (70 + 70 + 20) = 180 – 160 =20 Y = 20

19. X = 50

20. Practice
p. 382 #3, 4, 6, 7, 11, 13, 14, 17