CC Geometry 10H
Aim #2: How do we measure a central angle and an inscribed angle of a circle?
Do Now:
1) Using circle X below, fill in the circles with <, > or =. Then, write two examples
based on circle X.
B
a) Semi-circle:
0
180
0
Ex. _____
_____
b) Major arc:
180
Ex. _____
_____
c) Minor arc:
180
0
Ex. _____
_____
X
A
C
D
2) State the term that best describes each using circle O:
g) ACB
a) OA
C
b) AD
B
h) BC
c) BC
O
i) BAD
A
d) BC
D
j) D
E
e) DE
f) AB
A
A central angle is an angle whose sides are
____________ and its vertex is the
________________ of the circle.
O
B
Theorem: The measure of a CENTRAL ANGLE is equal to the measure of its
intercepted arc.
A
B
O
C
An inscribed angle is an angle whose sides are
_____________ and whose vertex is on the
___________.
Theorem: The measure of an INSCRIBED ANGLE is equal to one-half the
measure of its intercepted arc.
B
Corollary: An angle inscribed in a semi-circle is a right angle.
A
O
C
Corollary: Inscribed angles that intercept the same arc are congruent.
1
2
≮s 1 and 2 intercept _______, therefore they are congruent.
A
B
Theorem: In a circle, opposite angles of an inscribed
quadrilateral are ______________.
Algebraic proof:
m≮1 =
m_____
m≮2 =
m_____
m≮1 + m≮2 =
(
B
1
C
A
2
D
+
) or
______ = ______
Practice.
1) Solve for the missing variable(s):
a)
b)
c)
2) Quadrilateral ABCD is inscribed in a circle. If m≮A = 92 and m≮B = 71, find
m≮C and m≮D.
3) AB is a diameter of the circle shown. The radius is 12.5 cm, and AC = 7 cm.
a. Find m≮C. _______
b. Find CB.
4) Find the measure of angle x.
a)
0
m≮D = 25
d)
0
m≮B = 64
5) Find m≮VZW
b)
e)
0
m≮D = 15
c)
f)
0
m≮D = 19
6) Find m≮ADC
7) In the circle shown, BC is a diameter with center A.
a. Find m≮DAB.
b. Find m≮BAE.
E
c. Find m≮DAE.
D
C
0
8) Toby says that ΔBEA is a right triangle and m≮BEA=90 . Is he correct?
Justify your answer.
9) In the figure below, AB is the diameter of a circle of radius 17 miles.
C
If BC = 30, what is AC?
A
B
10) In the figure below, O is the center of the circle and AD is the diameter.
a) Find m≮AOB.
b) If m≮AOB:m≮COD = 4:3, what is m≮BOC?
Name: _____________________
CC Geometry 10H
HW#2
Date: _________________
For Problems #1-6, Find the value of x (not all angles are drawn to scale):
1)
2)
3)
4)
5)
6)
o
7) AB and CD are diameters of circle O. If m≮AOC = 110 , find the measures of
the following arcs:
D
B
A
O
C
a) AD
e) DBC
b) DB
f) DAB
c) BC
g) ACD
d) CA
T
8)
o
RS is a diameter of circle O, m≮ROT = 46 . Find:
R
a) mSRT
c) mTS
b) mRT
d) mRST
O
S
9) In circle O, mAE : mEC : mBC : mBD : mAD = 2:3:4:5:6. Find:
a) mAE
C
d) m≮EOC
E
A
b) mBC
e) mEBD
c) m≮AOD
f) mEDB
B
O
D
x
10) Given diagram on the right, find the values of x, y and z.
z
1200
y
Mixed Review:
1) In the diagram, ΔABC ~ ΔDEF. AC =6, AB = BC= 12, and DF = 8. Find the
perimeter of ΔDEF.
0
2) In the diagram below, AC ≅ DC ≅ DB. If the m≮ACD = 48 , find m≮B.