6.5 Other Angle Relationships in Circles Geometry.
-
Upload
ophelia-sullivan -
Category
Documents
-
view
217 -
download
0
Transcript of 6.5 Other Angle Relationships in Circles Geometry.
![Page 1: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/1.jpg)
6.5 Other Angle Relationships in Circles
Geometry
![Page 2: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/2.jpg)
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.
![Page 3: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/3.jpg)
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
![Page 4: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/4.jpg)
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
![Page 5: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/5.jpg)
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
![Page 6: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/6.jpg)
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
![Page 7: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/7.jpg)
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
![Page 8: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/8.jpg)
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
![Page 9: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/9.jpg)
Inside the circle
![Page 10: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/10.jpg)
Outside the circle
![Page 11: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/11.jpg)
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.
![Page 12: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/12.jpg)
Theorem 6.13
• 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
![Page 13: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/13.jpg)
Theorem 6.14
• 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
![Page 14: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/14.jpg)
Theorem 6.14
• 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
![Page 15: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/15.jpg)
Theorem 6.14
• 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
![Page 16: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/16.jpg)
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°
![Page 17: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/17.jpg)
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°
![Page 18: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/18.jpg)
P
N
M
L
Ex. 4: Using Theorem 6.14
• 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
![Page 19: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/19.jpg)
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
![Page 20: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/20.jpg)
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
![Page 21: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/21.jpg)
• 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
![Page 22: 6.5 Other Angle Relationships in Circles Geometry.](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfd71a28abf838cae759/html5/thumbnails/22.jpg)
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°.