9.7. Angles On and Inside a Circle http://www.ck12.org
- Draw
⊙
Awith chordBCand tangent line
←→
EDwith point of tangencyC.
- Draw in central angle^6 CAB. Then, using your protractor, findm^6 CABandm^6 BCE.
- FindmBĈ(the minor arc). How does the measure of this arc relate tom^6 BCE?
This investigation proves the Chord/Tangent Angle Theorem.
Chord/Tangent Angle Theorem:The measure of an angle formed by a chord and a tangent that intersect on the
circle is half the measure of the intercepted arc.
From the Chord/Tangent Angle Theorem, we now know that there are two types of angles that are half the measure
of the intercepted arc; an inscribed angle and an angle formed by a chord and a tangent. Therefore,any angle with
its vertex on a circle will be half the measure of the intercepted arc.
An angle is considered inside
Investigation: Find the Measure of an Angle
Tools Needed: pencil, paper, compass, ruler, protractor, colored pencils (optional)
- Draw
⊙
Awith chordBCandDE. Label the point of intersectionP. - Draw central angles^6 DABand^6 CAE. Use colored pencils, if desired.
- Using your protractor, findm^6 DPB,m^6 DAB, andm^6 CAE. What ismDB̂andmCÊ?
- FindmDB̂+ 2 mCÊ.
- What do you notice?
Intersecting Chords Angle Theorem:The measure of the angle formed by two chords that intersectinsidea circle
is the average of the measure of the intercepted arcs.
Inthepicturebelow:
m^6 SV R=
1
2
(
m̂SR+mT Q̂
)
=
mSR̂+mT Q̂
2
=m^6 T V Q
m^6 SV T=
1
2
(
mST̂+mRQ̂
)
=
mST̂+mRQ̂
2
=m^6 RV Q
Example A
FindmAEB̂
Use the Chord/Tangent Angle Theorem.
mAEB̂= 2 ·m^6 DAB= 2 · 133 ◦= 266 ◦
Example B
Findm^6 BAD.
Use the Chord/Tangent Angle Theorem.
m^6 BAD=^12 mAB̂=^12 · 124 ◦= 62 ◦
Example C
Finda,b, andc.