Theorem 7-10 The measure of an inscribed angle is equal to half the measure of its intercepted
arc.
Given: ∠ABC inscribed in circle P
Prove: m ∠ABC = m
We have three very useful corollaries to this theorem.
Corollary 1: An angle inscribed in a semicircle is a right angle.
Corollary 2: The opposite angles of an inscribed quadrilateral are supplementary.
Corollary 3: If two angles intercept the same or equal arcs, the angles are equal.
Model 2:
Model 1:
If ABCD is inscribed in circle O, then ∠A
If
is a diameter, then ∠RST is a right
and ∠C are supplementary and ∠B and ∠D
angle.
are supplementary.
Model 3:
m∠1 = m∠ 2 = m∠ 3.
If a diameter is perpendicular to
Theorem
a chord, then it bisects the chord
7-7
and its two
arcs.
The following summary of angle and arc
relationships should be useful for learning
and review.
Given:
m
Prove:
AC = CB
Inscribed angle = ½ intercepted arc
m
=m
m
m
=m
Central Angle = intercepted arc
1=m
1= ½ m
The measure of the angle formed
Theorem by two secants intersecting
outside the circle equals half the
7-13
difference of the intercepted arcs.
Given: Secant rays
Prove:
1 = ½ (m
and
-m
Angle formed by tangent and secant with
vertex on circle = ½ intercepted arc
)
m
1=½m
Angle formed by two secants with vertex
inside
circle = ½ sum of intercepted arcs
m
1 = ½ (m
+m
)
If two chords intersect in a circle,
the product of the lengths of the
Theorem
segments of one chord is equal to
7-16
the product of the lengths of the
other chord.
Given: Chords
and
intersect at P
Prove: AP·PC = BP ·
Angle formed by two secants with vertex
outside the
circle = ½ difference of intercepted arcs
m
1 = ½ (m
-m
PD
)
Angle formed by secant and tangent with
vertex outside the
circle = ½ difference of intercepted arcs
m
1 = ½(m
-m
If two secants intersect at a point
outside a circle, the length of one
Theorem secant times the length of its
external part is equal to the
7-17
length of the other secant times
the length of its external part.
Given: Secants
,
intersect at C
)
Prove: AC · BC = DC · EC
Angle formed by two tangents with vertex
outside the
circle = ½ difference of intercepted arcs
m
1 = ½ (m
-m
)
If a tangent segment and a secant
segment intersect outside a circle,
the length of the tangent segment
Theorem
is the geometric mean between
7-18
the secant segment and the
external part of the secant
segment.
Given: secants
P.
Prove:
, tangent
intersect at