Angles Related to a
Circle
Lesson 10.5

Inscribed Angle: an angle whose vertex is on a
circle and whose sides are determined by two
chords.

Tangent-Chord Angle: Angle whose vertex is
on a circle whose sides are determined by a
tangent and a chord that intersects at the
tangent’s point of contact.

Theorem 86: The measure of an inscribed
angle or a tangent-chord angle (vertex on circle)
is ½ the measure of its intercepted arc.
Angles with Vertices on a Circle
Angles with Vertices Inside, but
NOT at the Center of, a Circle.
Definition: A chord-chord
angle is an angle formed by two
chords that intersect inside a
circle but not at the center.
Theorem 87: The measure of
a chord-chord angle is one-half
the sum of the measures of the
arcs intercepted by the chordchord angle and its vertical
angle.
x = ½ (88 + 27)
½ a = 65
x = 57.5º
a = 130
½ (21 + y) = 72
21 + y = 144
y = 123º
Find y.
1. Find mBEC.
2. mBEC = ½ (29 + 47)
3. mBEC = 38º
4. y = 180 – mBEC
5. y = 180 – 38 = 142º
Part 2 of Section 10.5…
Angles with Vertices Outside a Circle
Three types of angles…
1. A secant-secant
angle is an angle whose
vertex is outside a circle
and whose sides are
determined by two
secants.
Angles with Vertices Outside a Circle
2. A secant-tangent
angle is an angle
whose vertex is outside
a circle and whose sides
are determined by a
secant and a tangent.
Angles with Vertices Outside a Circle
3. A tangent-tangent
angle is an angle whose
vertex is outside a circle
and whose sides are
determined by two
tangents.
Theorem 88: The measure of a secant-secant
angle, a secant-tangent angle, or a tangenttangent angle (vertex outside a circle) is ½ the
difference of the measures of the intercepted
arcs.
y = ½ (57 – 31)
½ (125 – z) = 32
y = ½(26)
125 – z = 64
y = 13
z = 61
1. First find the measure of arc EA.
2. m of arc AEB = 180 so
arc EA = 180 – (104 + 20) = 56
3. .
4. mC = ½ (56 – 20)
5. mC = 18
½ (x + y) = 65 and
½ (x – y ) = 24
x + y = 130
x – y = 48
x + y = 130
x – y = 48
2x = 178
x = 89
and
89 + y = 130
y = 41