CHAPTER 10.4 & 10.5
10.4 USING INSCRIBED ANGLES AND
POLYGONS
Inscribed angle- an angle whose vertex is on a
circle and whose sides contain chords of the
circle.
 Intercepted arc- an arc that lies in the interior of
an inscribed angle and has endpoints on the
angle.

Intercepted arc
Inscribed angle

Measure of an inscribed angle theorem- The
measure of an inscribed angle is one half the
measure of its intercepted arc.
A
1
mADB = mAB D
2
C
B

Example
A
D
B
50 C
35
E
Find
a. mD =
b. mAB =
EXAMPLE
A
52
D
E
B
Find
a. mDEB =
b. mDB =
C. DAB 
THEOREM

If 2 inscribed angles of a circle intercept the same
arc, then the angles are congruent.


Inscribed polygon- a polygon in which all the
vertices lie on the circle
Circumscribed circle- the circle that contains the
vertices of the polygon
THEOREM

A quadrilateral can be inscribed in a circle if and
only if its opposite angles are supplementary
B
A
mA + mC = 180
mB + mD = 180
D
C
EXAMPLE

Find x and y.
B
x
A
y
100
60
D
C
EXAMPLE

Find x and y.
A
17y
5x
D
B
7x
19y
C
THEOREM

If a right triangle is inscribed in a circle, then the
hypotenuse is a diameter of the circle.
Conversely, if one side of an inscribed triangle is
a diameter of the circle, then the triangle is a
right triangle and the angle opposite the
diameter is the right angle.
A
B
D
C
10.5 APPLYING OTHER ANGLE
RELATIONSHIPS IN CIRCLES
Theorem 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.

B
C
1
m1=
mAB
2
1
m2 = mACB
2
2 1
A
INTERSECTING LINES AND CIRCLES
on the circle
inside the circle
outside the circle
ANGLES IN THE CIRCLE THEOREM

If the chords intersect inside 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
D
1
m1 = mDC + mAB
2
A
1
2
m2 =
B
C
1
mAD + mBC
2
ANGLES OUTSIDE THE CIRCLE THEOREM

If a tangent and a secant, 2 tangents, or 2
secants intersect outside a circle, then the
measure of the angle formed is one half the
difference of the measures of the intercepted arcs.
1
2
3