warm up. p a b case i: central angle: vertex is at the center

Warm up

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Warm up

Central ANGLE = ARC

Case I: Central Angle: Vertex is AT the center

Case II: Inscribed Angle: Vertex is ON circle

ANGLE = ARC

2ANGLE

ARCANGLE

ARC

Page 4: Warm up. P A B Case I: Central Angle: Vertex is AT the center

Intercepted Arc:

• An arc whose endpoints are the two points of intersection of an angle with the circle and all points that lie within the angle.

Page 5: Warm up. P A B Case I: Central Angle: Vertex is AT the center

ArcdIntercepteAngleInscribed

160

The arc is twice as big as the angle!!

Page 6: Warm up. P A B Case I: Central Angle: Vertex is AT the center

120

Find the value of x and y

Page 7: Warm up. P A B Case I: Central Angle: Vertex is AT the center

Examples1. If mJK = 80 and JMK = 2x – 4, find x.

2. If mMKS = 56, find m MS.

x = 22

112

Page 8: Warm up. P A B Case I: Central Angle: Vertex is AT the center

72˚

If two inscribed angles intercept the same arc, then they are congruent.

Find the measure of DOG and DIG D

Page 9: Warm up. P A B Case I: Central Angle: Vertex is AT the center

If all the vertices of a polygon touch the edge of the circle, the polygon is INSCRIBED and the circle is CIRCUMSCRIBED.

Page 10: Warm up. P A B Case I: Central Angle: Vertex is AT the center

Circumscribed Circle• The circumscribed circle (or

circumcircle) of a polygon is a circle which passes through all the vertices of the polygon.

• The center of this circle is called the circumcenter and its radius is called the circumradius.

Page 11: Warm up. P A B Case I: Central Angle: Vertex is AT the center

a quadrilateral inscribed in a circle: opposite angles are

supplementary.A B

180 CmAm180 DmBm

Page 12: Warm up. P A B Case I: Central Angle: Vertex is AT the center

If a right triangle is inscribed in a circle then the hypotenuse is the diameter of the circle.

diameter

Page 13: Warm up. P A B Case I: Central Angle: Vertex is AT the center

Example 3

In J, m3 = 5x and m 4 = 2x + 9.Find the value of x.

x = 3

Page 14: Warm up. P A B Case I: Central Angle: Vertex is AT the center

4x – 14 = 90

Example 4

In K, GH is a diameter and mGNH = 4x – 14. Find the value of x.

x = 26

Bonus: What type of triangle is this? Why?

Page 15: Warm up. P A B Case I: Central Angle: Vertex is AT the center

110

110 + y =180y = 70

z + 85 = 180z = 95

Example 5 Find y and z.

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