10.4 Other Angle Relationships in Circles
Geometry
Fall 2012
Objectives/Assignment
• Use angles formed by tangents and chords to solve problems in geometry.
• Use angles formed by lines that intersect a circle to solve problems.
• Assignment: pp. 624-625 #2-35
Using Tangents and Chords
• You know that measure of an angle inscribed in a circle is half the measure of its intercepted arc. This is true even if one side of the angle is tangent to the circle.
C
B
A
D
ABmADB = ½m
Theorem 10.12• If a tangent and a
chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. 2
1
A
B
C
m1= ½m
m2= ½m
ABBCA
Ex. 1: Finding Angle and Arc Measures
• Line m is tangent to the circle. Find the measure of the red angle or arc.
• Solution:
m1= ½
m1= ½ (150°)
m1= 75°
m
1
B
A
150°AB
Ex. 1: Finding Angle and Arc Measures
• Line m is tangent to the circle. Find the measure of the red angle or arc.
• Solution:
m = 2(130°)
m = 260°
RSP
130°
RSP
RP
S
Ex. 2: Finding an Angle Measure• In the diagram below,
is tangent to the circle. Find mCBD
• Solution:
mCBD = ½ m
5x = ½(9x + 20)
10x = 9x +20
x = 20
mCBD = 5(20°) = 100°
DAB
(9x + 20)°
BC
B
C
A
5x°
D
Lines Intersecting Inside or Outside a Circle• If two lines intersect a circle, there
are three (3) places where the lines can intersect.
on the circle
Inside the circle
Outside the circle
Lines Intersecting
• You know how to find angle and arc measures when lines intersect
ON THE CIRCLE.
• You can use the following theorems to find the measures when the lines intersect
INSIDE or OUTSIDE the circle.
Theorem 10.13
• If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
2
1
B
A C
D
CDm1 = ½ m + m
AB
BCm2 = ½ m + m
AD
Theorem 10.14
• If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.
BCm1 = ½ m( - m )
AC
1
A
B
C
Theorem 10.14
• If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.
PQRm2 = ½ m( - m ) PR
2
R
P
Q
Theorem 10.14
• If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.
XYm3 = ½ m( - m ) WZ
Z
W
Y
X
3
Ex. 3: Finding the Measure of an Angle Formed by Two Chords
• Find the value of x
• Solution:
x° = ½ (m +m
x° = ½ (106° + 174°)
x = 140
PS
RQ
R
S
P
Q
Apply Theorem 10.13
Substitute values
Simplify
174°
106°
x°
Ex. 4: Using Theorem 10.14
• Find the value of x
Solution:
72° = ½ (200° - x°)
144 = 200 - x°
- 56 = -x
56 = x
Substitute values.
Subtract 200 from both sides.
Multiply each side by 2.
EDGmGHF = ½ m( - m )
GF Apply Theorem 10.14
Divide by -1 to eliminate negatives.
F
GH
E
D
200°
x°
72°
P
N
M
L
Ex. 4: Using Theorem 10.14
• Find the value of x
Solution:
= ½ (268 - 92)
= ½ (176)
= 88
Substitute values.
Multiply
Subtract
MLNmGHF = ½ m( - m )
MN Apply Theorem 10.14
x°92°
Because and make a whole circle, m =360°-92°=268°
MN
MLN
MLN
Ex. 5: Describing the View from Mount Rainier
• You are on top of Mount Rainier on a clear day. You are about 2.73 miles above sea level. Find the measure of the arc that represents the part of Earth you can see.
CD
Ex. 5: Describing the View from Mount Rainier
• You are on top of Mount Rainier on a clear day. You are about 2.73 miles above sea level. Find the measure of the arc that represents the part of Earth you can see.
CD
• and are tangent to the Earth. You can solve right ∆BCA to see that mCBA 87.9°. So, mCBD 175.8°. Let m = x° using Trig Ratios
Ex. 5: Describing the View from Mount Rainier
BCCD
BD
CD
175.8 ½[(360 – x) – x]
175.8 ½(360 – 2x)
175.8 180 – x
x 4.2
Apply Theorem 10.14.
Simplify.
Distributive Property.
Solve for x.
From the peak, you can see an arc about 4°.
Reminder:
• Quiz after section 10.5
• Deficiencies go out the week of April 25-28; so all work is due no later than the end of this week.
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