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Day 5
Inscribed
Angles
Inscribed Angles


An inscribed angle has its vertex on the circle
and sides are chords of a circle.
An intercepted arc is the arc that lies in the
interior of an inscribed angle and has endpoint
on the angle
inscribed angle
intercepted arc
VERY IMPORTANT:
 FACT: the measure of
an inscribed angle is half
the measure of the arc it
intercepts.
60

120
Theorem

If two inscribed angles of a circle intercept the
same arc, then the angles are congruent
B
A
C

D
A  B
Since both angles intercept arc CD
Inscribed Polygon

If all vertices of a polygon lie on a circle, the
polygon is inscribed in the circle and the circle
is circumscribed about the polygon
B
A
C

E
D
Pentagon ABCDE is inscribed in the circle
Theorem:
 A right triangle is inscribed in a circle if and
only if the hypotenuse is a diameter of the
circle.
Theorem:

A quadrilateral can be inscribed in a circle if
and only if its opposite angles are
supplementary.
x
w
y
z
w  y  180
o
o
0
x  z  180
o
o
0
Let’s sum it all up: 4 circle properties we
will use today in our examples:
1. Inscribed angle = ½ (intercepted arc) OR
Intercepted arc = 2(inscribed angle)
2. If two inscribed angles intercept the same arc,
angles are congruent!
3. A right triangle is inscribed in a circle if and
only if the hypotenuse is the diameter.
4. A quadrilateral can be inscribed in a circle if
and only if opposite angles are supplementary.
Ex. 1: Find the measure of QRS
R
45
S

90
Q
mQRS  1 / 2(90)
mQRS  45
Ex. 2: Find the measure of ∠𝐴𝐷𝐵.
B
C  80
D
80
A
BA  80
Central angle = intercepted arc
1
o
o
ADB  (80)  40
2
Inscribed angle = ½(intercepted arc)
Ex. 3: Find the measure of x.
S
256
2(7 x  16)  256

Q
(7 x  16)
14 x  32  256
14 x  224
R
2(Inscribed angle) = intercepted arc
x  16
Ex. 4: Find x and y.
D
45o
B
yo
C
E
xo
40o
A
Both angles intercept the same arc so angles
are congruent!!
y  40
o
y  45
o
Ex. 5: Find x and y.
x
20
y
One side of triangle is
diameter right triangle!!
x = 90o
y = 180o – 90o – 20o
y = 70o
Ex. 6: Find x and y.
(15𝑦 + 3)𝑜
(8𝑥 + 2)𝑜
(7𝑥)𝑜
10(𝑦 + 4)𝑜
8𝑥 + 2 + 7𝑥 = 180
15𝑥 = 178
178
𝑥 =
15
A quadrilateral can be inscribed in
a circle if and only if its opposite
angles are supplementary.
15𝑦 + 3 + 10𝑦 + 40 = 180
25𝑦 = 137
137
𝑥 =
25
Ex. 7: Decide whether a circle can be
circumscribed about the quadrilateral.
96𝑜
116𝑜
84𝑜
64𝑜
A quadrilateral can be inscribed in
a circle if and only if its opposite
angles are supplementary.
360 − 116 − 96 − 64 = 84𝑜
116 + 64 = 180𝑜
96 + 84 = 180𝑜
Yes, since opposite angles supplementary