Section 10.4 Inscribed Angles
Definition: An inscribed angle is an angle whose vertex is on the circle and
whose sides contain chords of the circle.
Ex.: Draw a  with inscribed
s 1 and 2.
Theorem: The measure of an inscribed angle is half the measure of its
intercepted arc.
Ex.:
B
1. If BC = 80 , then
A
C
2. If
BAC = _____°.
BAC = 35°, then BC = _____°.
Theorem: If 2 inscribed angles of a circle intercept the same arc, then the 2
angles are congruent.
Ex.:
A
B
1
2
1. What arc does
1 intercept? ______
2. What arc does
2 intercept? ______
3. If
2 = 20 , then AB = ____° and
4. If AB = 34°, then
1 = _____° and
1 = _____°.
2 = _____°.
Definition: A circle is inscribed in a polygon if each side of the polygon is
tangent to the circle.
Ex: Is the circle inscribed in ∆ABC?
1.)
2.)
A
A
C
C
B
B
Definition: A circle is circumscribed about a polygon if each vertex of the
polygon lies on the circle.
A
o
B
C
Ex 1: Is O circumscribed about ABC? ________
Ex 2: Draw a circle that is not circumscribed about
∆XYZ.
X
Z
Y
Theorem: An angle inscribed in a semicircle is a right angle.
Ex.:
A
1. If AB is a diameter,
C
B
C must equal _____°.
2. Draw a circle that has PQ as a diameter.
Draw three angles inscribed in one of its
semicircles.
3. The measure of each inscribed is _____°.
Theorem: If a quadrilateral is inscribed in a circle, then its opposite angles are
supplementary.
Ex.:
1. Draw a circle with inscribed quadrilateral ABCD.
2. If
A = 50 , then
C = _____ .
3. If B = x2 + 20 and D = 9x – 2,
find x and m B. Show algebra.