Circles: Special Angles and Special Segments

Vertex ON a Circle Secant and Tangent Two Secants

Vertex Inside a Circle Two Secants

Vertex Outside a Circle Two Tangents Two Secants Secant & Tangent

Vertex ON a Circle < measure Find the missing measure 110 35 xx yy

Vertex ON a Circle < measure Angle = ½ arc 110 35 55 70

Vertex Inside a Circle < measure Find the missing measure 90 xx 30

Vertex Inside a Circle < measure Angle =big + small 90 60 30 2

Vertex Inside a Circle < measure Find the missing measure 160 yy 72

Vertex Inside a Circle < measure Find supplement first. 160 yy 72 116

Vertex Inside a Circle < measure Then subtract from 180 160 64 72 116

Find the missing measure Vertex Outside a Circle < measure xx 216 120 yy 22.5 15 zz 10 50

Angle =big - small Vertex Outside a Circle < measure 48 216 120 30 22.5 15 60 10 50 2

Special segments C X A B

Tangent Segment XA C X A B

Secant Segment XB C X A B

External Secant Segment XC C X A B

Chord CB C X A B

Tangents Given XA and XB are tangents, what can you conclude about the following diagram? X A B P

Tangents Given XA and XB are tangents, what can you conclude about the following diagram? X A B P XA  XB? Prove it!

Tangents What do you do to prove something? X A B P Make  s

Tangents What do you know about PA & PB? X A B P

Tangents What do you know about PA & PB? X A B P PA  PB All radii of a circle are 

Tangents What do you know about PX? X A B P

Tangents What do you know about PX? X A B P PX  PX Reflexive

Tangents What do you know about the  s? X A B P

Tangents What do you know about the  s? X A B P They are right  s Radius | tangents

Tangents What else do you know about the  s? X A B P

Tangents What do you know about the  s? X A B P  s  HL Thm

Tangents What do you know about XA & XB? X A B P

Tangents What do you know about XA & XB? X A B P XA  XB CPCTC

Secants AX  CX = BX  DX X A B P C D

Secants AX  CX = BX  DX X A B P Why does this work? C D

Secants AA similarity X A B P Angles intercept the same arc. C D O Therefore the s are proportional.

Secant & Tangent AX  CX = (BX) 2 X A B P C

Chords AX  XC = BX  XD X A C D B