Download Secants, Tangents and Angle Measures

Secants, Tangents and Angle Measures
• Find measures of angles formed by lines intersecting on or
inside a circle.
• Find measures of angles formed by lines intersecting
outside the circle.
Water droplet on a CD
INTERSECTIONS ON OR INSIDE A CIRCLE
S
B
F
E
D
A line that intersects a circle in exactly two points is called
a secant. In the figure above, SF and EF are secants of
the circle.
When two secants intersect inside a circle, the angles
formed are related to the arcs they intercept.
Theorem
If two secants intersect in the interior of a circle, then the
measure of the angle formed is one half the sum of the
measure of the arcs intercepted by the angle and its
vertical angle.
A
1
2
D
B
C
Examples:
1
m1  (mAC  mBD)
2
1
m2  (mAD  mBC )
2
Example 1
Secant-Secant Angle
Find m2 if mBC = 30
and mAD = 20
20°
A
D
2
B 30°
1
C
E
Example 2
Secant-Secant Angle
F
Find m4 if mFG = 88
and mEH = 76
88°
E
G
4
3
76°
H
A secant can also intercept a tangent at the point of
tangency.
Each angle formed has a measure half that of the arc it
intercepts.
1
mABC  mBC
2
1
mDBC  mBEC
2
B
A
C
D
E
Theorem
If a secant and a tangent intersect at the point of tangency,
then the measure of each angle formed is one half the
measure of the intercepted arc.
1
mABC  mBC
2
1
mDBC  mBEC
2
B
A
C
D
E
Example 3
Secant-Tangent Angle
Find the mABC if mAB = 102°
D
A
102°
B
C
Example 4
Secant-Tangent Angle
Find the mRPS if mPT = 114°
and mTS = 136°
R
P
Q
S
114°
B
T
136°
INTERSECTIONS OUTSIDE A CIRCLE
Secants and tangents can also intersect outside a circle.
The measure of the angle formed also involves half the
measure of the arcs they intercept.
Theorem
If two secants, a secant and a tangent, or two tangents
intersect in the exterior of a circle, then the measure of the
angle formed is one-half the positive difference of the
measures of the intercepted arcs.
Two Secants
Secant-Tangent
Two Tangents
D
D
C
E
B
B
D
A
1
mA  (mDE  mBC )
2
B
C
C
A
1
mA  (mDE  mBC )
2
A
1
mA  (mBDC  mBC )
2
Example 5
Secant-Secant Angle
Find x
x°
120°
50°
Example 6
Secant-Secant Angle
Find x
62°
141°
x°
Example 7
Secant-Secant Angle
Find x
11°
1
11  [(360  x)  x]
2
22  360  2 x
 338  2 x
169  x
x°