9.8. Angles Outside a Circle http://www.ck12.org
- Draw in all the central angles:^6 GAH,^6 EAF,^6 MBN,^6 RCT,^6 RCS. Then, find the measures of each of these
angles using your protractor. Use color to differentiate. - Findm^6 EDF,m^6 MLN, andm^6 RQS.
- FindmEF̂− 2 m̂GH,mMPN̂ 2 −mMN̂, andm̂RS− 2 mRT̂. What do you notice?
Outside Angle Theorem:The measure of an angle formed by two secants, two tangents, or a secant and a tangent
drawn from a point outside the circle is equal to half the difference of the measures of the intercepted arcs.
Example A
Find the value ofx. You may assume lines that look tangent, are.
Set up an equation using the Outside Angle Theorem.
( 5 x+ 10 )◦−( 3 x+ 4 )◦
2= 30 ◦
( 5 x+ 10 )◦−( 3 x+ 4 )◦= 60 ◦
5 x+ 10 ◦− 3 x− 4 ◦= 60 ◦
2 x+ 6 ◦= 60 ◦
2 x= 54 ◦
x= 27 ◦Example B
Find the value ofx.
x=^120
◦− 32 ◦
2 =
88 ◦
2 =^44◦.
Example C
Find the value ofx.
First note that the missing arc by anglexmeasures 32◦because the complete circle must make 360◦. Then,x=
141 ◦− 32 ◦
2 =
109 ◦
2 =^54.^5◦.
Watch this video for help with the Examples above.
MEDIA
Click image to the left for more content.CK-12 Foundation: Chapter9AnglesOutsideaCircleB
Concept Problem Revisited
If 178◦of the Earth is exposed to the sun, then the angle at which the sun’s rays hit the Earth is 2◦. From the Outside
Angle Theorem, these two angles are supplementary. From this, we also know that the other 182◦of the Earth is not
exposed to sunlight and it is probably night time.