Geometry
Unit 10 Day 2
Name_________________________
Warm Up
Find x and y.
1.
x
2.
40
120
70
x
y
y
3. In the diagram, S is a point of tangency. Find the radius.
T
r
36
r
S
4. In
R
48
C, DA is tangent at A and DB is tangent at B. Find x.
A
25
D
C
6x - 8
B
Finding Arc Measures & Using Inscribed Angles and Polygons
central angle –
A
minor arc –
C
major arc –
semicircle –
D
B
Measuring Arcs
The measure of an arc is EQUAL to the measure of its corresponding central angle
Example 1
Find the measure of each arc of circle P, where RT is a diameter.
R
a. RS
P
b. RTS
c. RST
110
T
S
adjacent arcs –
A
Arc Addition Postulate –
The measure of an arc formed by two adjacent arcs
is the sum of the measure of the two arcs.
C
B
Example 2
Identify the given arc as a minor arc, major arc, or semicircle. Then find the measure of the arc.
a. TQ
b. QRT
c. TQR
T
80
d. QS
e. TS
f. RST
120
60
S
Q
R
inscribed angle –
intercepted arc –
Theorem –
The measure of an inscribed angle is one half the measure of its intercepted arc.
Q
Example 3
Find the indicated measure in circle P.
a. mT
P
b. QR
R
48
T
Example 4
50
S
Find the measure of RS and mSTR. What do you notice about STR and RUS?
T
S
R
31
U
Theorem –
If two inscribed angles intercept the same arc, then the angles are congruent.
Example 5
Find the measure indicated.
a. FGH
b. TV
H
T
X
U
38
D
G
c. WXZ
Y
72
90
F
Z
V
W
A polygon is an inscribed polygon if all of its vertices lie on the circle. The circle that contains the
vertices is a circumscribed circle.
Theorem –
If a right triangle is inscribed in a circle, then the hypotenuse is the diameter.
Theorem –
A quadrilateral can be inscribed in a circle iff
its opposite angles are supplementary.
Example 6
Find the value of each variable.
a.
80
b.
2a
2b
y
4b
75
x
2a