10.4 other angle relationships in circles
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10.4 Other Angle Relationships in Circles. Learning Target. I can use theorems about tangents, chords and secants to solve unknown measure of arcs and angles. Review on inscribe angles. - PowerPoint PPT PresentationTRANSCRIPT
10.4 Other Angle Relationships in Circles
Learning Target
• I can use theorems about tangents, chords and secants to solve unknown measure of arcs and angles .
Review on inscribe angles
• 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
n
x
Angle x = ½ n
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
n
x
Angle x = ½ n
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
Angle x = ½ n
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
Angle x = ½ n
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
Angle x = ½ n
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
n
x
f
Angle x = ½ ( f + n)
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
nx
f
Angle x = ½ ( f - n )
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
Angle x = ½ ( f - n )
fnx
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
fnx
Angle x = ½ ( f - n )
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°Angle x = ½ (f +n)
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°
Angle x = ½ (f – n)
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
Angle x = ½ ( f – n )
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°.
Reminders:
• Pair-Share:
Work on page. 624-625 #2-35
• Refer to the summary sheet “Angles Related to Circles” to identify what formula to use.
• Quiz on Friday about Trigonometry and Circles.