Download Inscribed Angle

12 .5
Inscribed Angles and Triangles
Recall: A Central Angle has the same
measure as the arc in intercepts.
.100°
An inscribed
100° angle is not
equal to the arc
it intercepts.
B
Chords
Inscribed Angle - An
angle in a circle whose
vertex is on the circle and
sides contain cords of the
circle.
A
C
Intercepted Arc
•Ex: AC
•Measured in degrees
• Ex: ABC
.
P
23°
46°
What do you notice
about the measures
of the central and the
inscribed angles?
An inscribed angle is always half of
the central angle with the same
intersected arc.
measure
of
= ½ (measure of central angle)
inscribed
angle
x°
(½)x°
I think of an inscribed angle as being
farther away from the arc, so it is
smaller.
Ex 1 - Find mDOG, mDUG, and
mDIG given that the measure of arc DG
is 50°:
mDOG = 50°
50°
G
since it is a central
D
angle
mDUG = ½ of 50°
= 25°
since it is an
O
inscribed angle
mDIG = 25°,
since it is also and
inscribed angle
I
U
Ex.1: Find the measure of angle b and arc
a given that mPQT = 60° and m TS = 30°.
mQ = ½m PT
P
a° = 120°
60 = ½m PT
120 = m PT
T
60°
30°
Q
mb = ½m PS
mb = ½(120 + 30)
mb = 75°
S
b°
R
Ex 2 – Find the mDOT
The arc
intercepted
by DFT is
double
mDFT
D
So, mDOT = 24°
O
?
F
24°
48°
T
Corollary 1 for Inscribed Angle
• Two inscribed angles that intercept the same
arc are congruent.
B
100°
50°
A
50°
D
C
Ex 3 - Find the mTAP, given that
OT = 80° and OP = 110°.
If you add TO and
OP, you get 190°.
O
80°
110°
T
P
A
Since TAP is
an inscribed
angle, divide
the arc in half
to get:
mTAP = 95°
Corollary 2 for Inscribed 
• An  inscribed in a semicircle is a right .
180°
90°
Ex 4 – Find mRAT given that RA
measures 70°.
TA is a
70°
A
55°
.
R
35°
T
Angles in a
triangle
add up to
180°
.
Q
diameter
since it
passes
through
the center.
TQA measures
180°, so
mTRA = 90°
Ex 5 – find x:
Because a side passes
through the center, the
angle opposite the
hypotenuse is 90°.
x²+(3x)² = 10²
10
1x²+ 9x² = 100
.
10x² = 100
3x
x
x² = 10
x = √10
Example:
• 2(68) = 136 = XW
• 80 + 136 + ? = 360
• ? = 144
Example:
•
•
•
•
•
•
•
•
•
•
2(32) = 64 = YZ
YZ + XY = 180
64 + 15x + 11 = 180
75 + 15x = 180
15x = 105
x=7
Or 90 + 32 = 122
180 – 122 = 58
15x + 11 = 116
x=7
Example:
• 2(4x–5) = CDN
• 8x – 10 + 6x + 10 + 150
=360
• 14x = 360 – 150
• 14x = 210
• x = 15
Example:
•
•
•
•
•
•
•
3x + 23 + 9x + 1 = 180
12x + 24 = 180
12x = 156
x = 13
3(13) + 23 = 62
TS = 62
Angle TRS = 31
Example:
•
•
•
•
•
•
•
•
•
YRS = 2 (2 + 49x)
4 + 98x + 49x + 2 + 31x – 2
4 + 178x = 360
178x = 356
x=2
2 + 49(2) = 100
Check: 100 + 200 + 60 = 360
YRS = 200 TY = 49(2)+2 =100
TS = 31(2) – 2 = 60
What have we learned??
• The measure of a
central angle = the
measure of its
intercepted arc.
• The measure of an
inscribed angle = ½ the
measure of its
intercepted arc.
50°
A
60°
120°
120°
100°
If the hypotenuse of an inscribed
triangle is the diameter of the circle,
then it is a right triangle.
.
Then you can use
the fact that angles
in a triangle add up
to 90°, and inscribed
angles are half of
their intercepted
arcs to find missing
angles and arcs.