Circle Theorems
Chapter 10
Section 10.1
(Tangents)
Theorem 10.1
–
A line is tangent if and only if it is perpendicular to the radius of the
circle.
Theorem 10.2
–
Tangent segments from a common external point are congruent.
Section 10.3
(Chords)
Theorem 10.3
–
Two minor arcs are congruent if and only if their chords are congruent.
PQ is perpendicular to line
m
, therefore line
m is tangent to circle P.
SR = QR
AB = CD
So
mAB = mCD
Theorem 10.4
–
If one chord is the perpendicular bisector of another
chord, then one of
the chords (the longe
st) is a diameter.
Theorem 10.5
–
If a diameter of a circle is perpendicular to a chord then the diameter
bisects the chord and its arc.
Theorem 10.6
–
In the same circle or in congruent circ
les,
two chords are congruent if
and
only if they are equidistant from the center.
AB is the perpe
ndicular bisector of CD, so AB is a
diameter of the circle.
QR is the perpendicular bisector of ST, so SW = TW
And SR = TR
EF = EG
so
AB = CD
Section 10.4
(Inscribed Angles)
Theorem 10.7
–
The measure if an inscribed angle is one half the measure if its
intercepted arc.
Theorem 10.8

If two inscribed angles
of a circle intercept the same arc, then the angles
are congruent.
Theorem 10.9
–
If a right triangle is inscribed in a circle, then the hypotenuse is the
diameter of the circle.
m< ADB = ½ mAB
m<ADB = m<ACB
Triangle ABC is a right triangle with AB as the
hypotenuse, so AB is the diameter of the circle.
Theorem 10.10
–
A quadrilateral can be
inscribed in a circle if and only if its opposite
angles are supplementary.
Section 10.5
(Inside, Outside, On)
Theorem 10.11
–
If a tangent and a chord intersect at a point
on
a circle, then the measure
of each angle formed is one half th
e measure of its intercepted arc.
Theorem 10.12
–
If two 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.
m<1 + m<2 = 180
°
m<1 = ½ mAB
m<2 = ½
mACB
m<1 = ½ (mDC + mAB)
m<2
–
=
봠⡭½C䅄)
=
Theor
em 10.13
–
If a tangent and a secant, two tangents, or two secants intersect
outside
the circle, then the measure of the angle formed is one half the
difference
of the measures
of the intercepted arcs.
Section 10.6
(Segment Lengths)
Theore
m 10.14
–
If two chords intersect in the interior of a circle, then the product of the
lengths of the segments of one chord is equal to the product of the lengths of the
segments of the other chord.
Theorem 10.15
–
If two secant segment
s share a common endpoint outside the circle,
then the product of the lengths of one secant segment and its external segment equal the
product of the other secant segment and its external segment.
m<1 = ½ (mPQR
–
=
m
SR)
=
EA • EB = ED • EC
=
EA • EB = EC • ED
=
Theorem 10.16
–
If a secant segment and a
tangent segment share an endpoint outside a
circle, then the product of the lengths of the secant segment and its external segment
equals the square of the length of the tangent segment.
EA² = EC • ED
=
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