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High School Geometry
Postulates,
Theorems, &
Corollaries
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Segments & Angles
Ruler Postulate - The points on a line can be matched one to one with the real numbers.
Segment Addition Postulate - If B is
between A and C, then AB+BC=AC.
ab
bc
a
b
c
ac
Protractor Postulate - Given AB and a point O on AB, all rays that can be drawn from O
can be put into a one-to-one correspondence with the real numbers from 0 to 180.
Angle Addition Postulate - If S is in the interior
of ∠PQR, then m∠PQS + m∠SQR = m∠PQR.
p
s
x°
(x+y)°
q
y°
r
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Points, Lines, & Planes
a
Two Point Postulate - Through any two points
there is exactly one line.
y
Line a is the only possible line
that can run through
points X and Y
a
b
Lines a and b intersect at
point X.
x
Line-Point Postulate - A line contains at least two points.
Line Intersection Postulate - If two lines
intersect, then they intersect in exactly
one point.
x
Three Point Postulate - Through any three noncollinear points there is exactly one plane
containing them.
Plane-Point Postulate - A plane contains at least three noncollinear points.
Plane-Line Postulate - If two points lie in a plane,
then the line containing those points also
lies in the plane.
a
Points X and Y lie in plane Q,
and line a contains
points X and Y,
so line a is also in plane Q
y
x
Q
Plane Intersection Postulate - If two planes intersect,
then they intersect in exactly one line.
a
Planes R and Q
intersect in line a.
R
Q
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Angle Relationships
Linear Pair Theorem - It two angles form a linear pair, then they are supplementary.
Congruent Supplements Theorem - If two angles
are supplementary to the same angle
(or to 2 congruent angles), then the two
angles are congruent.
Congruent Complements Theorem - If two angles
are complementary to the same angle (or to
2 congruent angles), then the two angles are
congruent.
Vertical Angles Theorem - If two angles are
vertical angles, then they are congruent.
1
2
3
2
1
1
2
4
3
3
If ∠’s 1 & 2 are supplementary
and
∠’s 2 & 3 are supplementary
then ∠ 1 ∠ 3
If ∠’s 1 & 2 are complementary
and
∠’s 2 & 3 are complementary
then ∠ 1 ∠ 3
∠’s 1 & 3 are vertical angles,
so ∠ 1 ∠ 3
∠’s 2 & 4 are vertical angles,
so ∠ 2 ∠ 4
Right Angles Congruent Theorem - All right angles are congruent
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Parallel Lines Cut by A Transversal
Corresponding Angles Postulate - If two parallel lines
are cut by a transversal, then the pairs of
corresponding angles are congruent.
1
3
1 2
Interior 3 54 6
7 8
Alternate Exterior Angles Theorem - If two parallel lines
are cut by a transversal, then the pairs of
alternate exterior angles are congruent.
Exterior1
Consecutive Interior Angles Theorem (Same-Side Interior Angles) If two parallel lines
are cut by a transversal, then the pairs of
same-side interior angles are supplementary
Exterior
3
2
4
5 6
7 8
Converse of the Alternate Interior Angles Theorem If two lines are cut by a transversal so that a pair
of alternate interior angles are congruent,
then the two lines are parallel.
1 2
Interior 3 54 6
7 8
Converse of the Alternate Exterior Angles Theorem If two lines are cut by a transversal so that a pair
of alternate exterior angles are congruent,
then the two lines are parallel.
Exterior1
Converse of the Consecutive Interior Angles Theorem (Same-Side Interior Angles) If two lines are
cut by a transversal so that a pair of
same-side interior angles are supplementary,
then the two lines are parallel.
Exterior
3
∠4
4
5 6
7 8
2
3
4
Interior
5 6
7 8
1
∠3
2
1
Converse of the Corresponding Angles Postulate If two lines are cut by a transversal so that a pair
of corresponding angles are congruent,
then the two lines are parallel.
∠2
4
5 6
7 8
Alternate Interior Angles Theorem - If two parallel lines
are cut by a transversal, then the pairs of
alternate interior angles are congruent.
3
∠ 5
∠ 6
∠ 3 ∠ 7
∠ 4 ∠ 8
∠1
2
∠1
∠2
1 2
Interior 3 54 6
7 8
∠ 8
∠ 7
∠ 4 & ∠ 6 are
supplementary
∠ 3 & ∠ 5 are
supplementary
If ∠ 1 ∠ 5, ∠ 2 ∠ 6,
∠3
∠ 7, or ∠ 4 ∠ 8
∠3
∠ 6 or ∠ 4 ∠ 5
∠1
∠ 8 or ∠ 2 ∠ 7
then,
the lines are parallel
If
then,
the lines are parallel
If
2
4
5 6
7 8
∠ 6
∠ 5
then,
the lines are parallel
If ∠ 4 & ∠ 6 are
supplementary
or
∠ 3 & ∠ 5 are
supplementary
then,
the lines are parallel
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Parallel & Perpendicular Lines
Perpendicular Transversal Theorem - In a plane,
if a transversal is perpendicular to one of
the two parallel lines, then it is
perpendicular to the other line.
Converse of the Perpendicular Transversal
Theorem - In a plane, if two lines are
perpendicular to the same line, then they
are parallel.
x
a
If line a is
perpendicular to line x,
then it is also
perpendicular to line y
a
If line x and line y are both
perpendicular to line a,
then
they are parallel
y
x
y
Transitive Property of Parallel Lines - If two lines are parallel to the same line, then
they are parallel to each other.
Slopes of Parallel Lines Postulate - In a coordinate plane, two non-vertical lines are
parallel if and only if they have the same slope. All vertical lines are parallel.
Slopes of Perpendicular Lines Postulate - In a coordinate plane, two non-vertical lines
are perpendicular if and only if the product of their slopes is -1. Vertical and
Horizontal lines are always perpendicular to each other.
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Congruent Triangles
Triangle Sum Theorem - The sum of the measures of the interior angles of a triangle is 180°
Corollary
to the Triangle Sum Theorem - The acute angles of a right triangle are
complementary
Corollary to the Triangle Sum Theorem - There can be at most one right or obtuse
angle in a triangle
Corollary to the Triangle Sum Theorem - The measure of each angle of an
equiangular triangle is 60°
Exterior Angle Theorem - The measure of an exterior
angle of a triangle is equal to the sum of the
measures of the two nonadjacent interior angles.
b
Third Angles Theorem - If two angles of one
triangle are congruent to two angles of a
second triangle, then the third angles of
the triangles are also congruent.
a
y°
(x+y)°
x°
z
x
If ∠ B ∠ Y and ∠ A ∠ X,
then
∠C
c
∠ Z
y
Isosceles Triangle Theorem - (Base Angles Theorem) If two sides of a triangle are
congruent, then the angles opposite the sides are congruent.
Corollary
to the Isosceles Triangle Theorem - If a triangle is equilateral, then
it is equiangular.
Converse of the Isosceles Triangle Theorem - (Converse of the Base Angles Theorem)
If two angles of a triangle are congruent, then the sides opposite them are
congruent.
Corollary
to the Converse of the Isosceles Triangle Theorem - If a triangle is
equiangular, then it is equilateral.
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Congruent Triangles
b
Side-Side-Side (SSS) Congruence Postulate - If three sides of
one triangle are congruent to three sides of a second
triangle, then the triangles are congruent.
a
z
c
b
Side-Angle-Side (SAS) Congruence Postulate - If two sides and
the included angle of one triangle are congruent to two
sides and the included angle of another triangle, then the
triangles are congruent.
a
c
a
a
x
y
z
c
b
Angle-Angle-Side (AAS) Congruence Theorem - If two angles
and a non-included side of one triangle are congruent to
the corresponding angles and non-included side of
another triangle, then the triangles are congruent.
y
z
b
Angle-Side-Angle (ASA) Congruence Postulate - If two angles
and the included side of one triangle are congruent to
the two angles and the included side of another triangle,
then the triangles are congruent.
x
x
y
z
c
x
y
Hypotenuse-Leg (HL) Congruence Theorem - If the hypotenuse
and a leg of a right triangle are congruent to the
hypotenuse and leg of another right triangle, then the
triangles are congruent.
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Relationships in Triangles
Perpendicular Bisector Theorem - If a point is
on the perpendicular bisector of a
segment, then it is equidistant from the
endpoints of the segment.
c
d
a
Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints
of a segment, then it is on the
a
perpendicular bisector of the segment.
Angle Bisector Theorem - If a point is on the
bisector of an angle, then it is equidistant
from the sides of the angle.
Point C is on the perpendicular
bisector of AB,
thus
b
AC=BC
Point C is equidistant from the
endpoints of AB,
thus
Point C lies on the
b
perpendicular bisector of AB
c
d
b
c
d
a
Converse of the Angle Bisector Theorem - If a
point in the interior of an angle is equidistant
from the sides of the angle, then it on the
bisector of the angle.
a
Point C is equidistant from the
sides of ∠BAD,
thus
Point C lies on the
angle bisector of∠BAD
c
d
Circumcenter Theorem - The circumcenter of a
triangle is equidistant from the vertices
of the triangle.
X is the Circumcenter of the
shown triangle,
thus
X is equidistant from the vertices
x
Incenter Theorem - The incenter of a triangle is
equidistant from the sides of the triangle.
Centroid Theorem - The centroid of a triangle is
located ⅔ of the distance from each
vertext to the midpoit of the opposite
side
a
Point C is on the angle
bisector of ∠BAD,
thus
BC=CD
X is the Incenter of the
shown triangle,
thus
X is equidistant from the sides
x
c
b
X is the Centroid of the
shown triangle,
thus
d
x
f
e
AX=
⅔AD ;
CX=
⅔CF ;
& EX=
⅔EB
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Relationships in Triangles
Triangle Midsegment Theorem - The segment
connecting the midpoints of two sides of a
triangle is parallel to the third side and is half
as long as that side.
Triangle Longer Side Theorem - If one side of a
triangle is longer than another side, then the
angle opposite the longer side is larger than
the angle opposite the shorter side.
Triangle Larger Angle Theorem - If one angle of a
triangle is larger than another angle, then the
side opposite the larger angle is longer than
the side opposite the smaller angle.
Exterior Angle Inequality Theorem - The measure
of an exterior angle of a triangle is greater
than the measure of either of its
corresponding remote interior angles.
a
AB= ½CD
b
AB || CD
c
d
a
AC > AB
4
6
thus,
b
∠B > ∠C
c
a
∠B > ∠C
thus,
b 65°
40°
AC > AB
c
a
m∠1 > m∠A
and
m∠1 > m∠B
1
b
a
Triangle Inequality Theorem - The sum of the
lengths of any two sides of a triangle is
greater than the length of the third side.
Hinge Theorem - If two sides of one triangle are
congruent to two sides of another triangle
and the included angles are not congruent,
then the longer third side is across from the
larger included angle.
Converse of the Hinge Theorem - If two sides of
one triangle are congruent to two sides of
another triangle and the third sides are not
congruent, then the larger included angle is
across from the longer third side.
b
AB+BC > AC
BC+AC > AB
AC+AB > BC
c
x
a
∠Y > ∠B
thus,
b 60°
c
y 70°
z
XZ > AC
x
a
XZ > AC
8
b
c
10
y
z
thus,
∠Y > ∠B
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Polygons
Polygon Interior Angles Sum Theorem - The sum of the
measures of the interior angles of a convex
polygon with n sides is (n-2)*180°.

b
a
Corollary to the Polygon Interior Angles Sum TheoremThe sum of the measures of the interior angles of
a quadrilateral is 360°.
Polygon Exterior Angles Sum Theorem - The sum of the
measures of the exterior angles of a convex
polygon, one angle at each vertex, is 360°.
m∠A + m∠B + m∠C + m∠D + m∠E
= (5-2)*180°
c
=540°
e
d
2
1
m∠1 + m∠2 + m∠3 + m∠4 + m∠5
= 360°
3
5
4
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Parallelograms
Parallelogram Opposite Sides Theorem If a quadrilateral is a parallelogram,
then its opposite sides are congruent.
Parallelogram Opposite Angles Theorem If a quadrilateral is a parallelogram,
then its opposite angles are congruent.
Parallelograms Consecutive Angles Theorem If a quadrilateral is a parallelogram,
then its consecutive angles are supplementary.
Parallelogram Diagonals Theorem If a quadrilateral is a parallelogram,
then its diagonals bisect each other.
a
AB CD
AC BD
c
b
∠A ∠D
∠B ∠C
c
d
a
x°
y°
b
x+y= 180°
c
y°
x° d
a
b
AX XD
CX XB
x
c
Converse of the Parallelogram Opposite Angles Theorem If both pairs of opposite angles of a
quadrilateral are congruent, then the
c
quadrilateral is a parallelogram.
Converse of the Consecutive Angles Theorem If an angle of a quadrilateral is supplementary to
both of its consecutive angles, then the
c
quadrilateral is a parallelogram.
Parallel and Congruent Sides Theorem If one pair of opposite sides of a quadrilateral
are both congruent and parallel, then the
quadrilateral is a parallelogram.
d
a
Converse of the Parallelogram Opposite Sides Theorem If both pairs of opposite sides of a quadrilateral
are congruent, then the quadrilateral is a
c
parallelogram.
Converse of the Parallel Diagonals Theorem If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a parallelogram.
b
d
a
b
If
AB CD & AC BD
then
ABCD is a parallelogram
d
a
b
If
∠A ∠D & ∠B ∠C
then
ABCD is a parallelogram
d
a
x°
y°
y°
b
then
ABCD is a parallelogram
x° d
a
b
If
AX XD & CX XB
x
c
If
x+y= 180°
then
ABCD is a parallelogram
d
a
b
If
AB CD and AB ||CD
c
d
then
ABCD is a parallelogram
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Rectangles, Rhombi, & Squares
“A quadrilateral is a Rectangle if and only if it has Four Right Angles”
a
b
Rectangle Diagonals Theorem - If a parallelogram is a
rectangle, then its diagonals are congruent.
Converse of the Rectangle Diagonals Theorem If the diagonals of a parallelogram are
congruent, then the parallelogram is a rectangle.
Parallelogram Right Angle Theorem - If one angle of a
parallelogram is a right angle, then the
parallelogram is a rectangle.
AD BC
d
b
c
a
If
AD BC
then
d ABCD is a Rectangle
b
c
a
∠A
If
is a right angle
then
ABCD
is
a Rectangle
d
c
“A quadrilateral is a Rhombus if and only if it has Four Congruent Sides”
Rhombus Diagonals Theorem - If a parallelogram is a
rhombus, then its diagonals are perpendicular.
Converse of the Rhombus Diagonals Theorem - If the
diagonals of a parallelogram are perpendicular,
then the parallelogram is a rhombus.
b
a
c
d
b
a
Rhombus Opposite Angles Theorem - If a parallelogram
is a rhombus, then each diagonal bisects a pair
a
of opposite angles.
If
c
d
b
c
Parallelogram Consecutive Sides Theorem - If one pair
of consecutive sides of a parallelogram are
congruent, then the parallelogram is a rhombus.
b
a
d
AC BD
then
ABCD is a Rhombus
Diagonal BD bisects
∠B & ∠D
Diagonal AC bisects
∠A & ∠C
d
Converse of the Opposite Angles Theorem - If one
diagonal of a parallelogram bisects a pair of
opposite angles, then the parallelogram is a
rhombus.
AC BD
If Diagonal AC
bisects
∠A & ∠C
c
then
ABCD is a Rhombus
b
a
If
c
d
AB  BC
then
ABCD is a Rhombus
“If a quadrilateral is both a Rectangle and a Rhombus, then it is a Square”
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Trapezoids & Kites
Isosceles Trapezoid Base Angles Theorem - If a
trapezoid is isosceles, then each pair of base
angles is congruent.
Converse of the Isosceles Trapezoid Base Angles
Theorem - If a trapezoid has a pair of
congruent base angles, then it is an isosceles
trapezoid.
Isosceles Trapezoid Diagonals Theorem - A
trapezoid is isosceles if and only if its
diagonals are congruent.
Trapezoid Midsegment Theorem - The midsegment
of a trapezoid is parallel to each base, and its
length is one-half the sum of the lengths of
the bases.
a
b
∠B
∠C ∠D
∠A
d
c
a
b
If
∠A
∠B or ∠C ∠D
then
c Trapezoid ABCD is isosceles
d
a
b
If
AC BD
then
Trapezoi
d
ABCD
is isosceles
c
d
a
b
x
AB || XY || DC
y
d
XY= ½(AB+DC)
c
b
Kite Diagonals Theorem - If a quadrilateral is a kite,
then its diagonals are perpendicular.
a
c
AC  BD
d
Kite Opposite Angles Theorem - If a quadrilateral is
a kite, then exactly one pair of opposite
a
angles are congruent.
b
c
∠B
∠D
d
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Similarity & Proportions
Perimeters of Similar Polygons Theorem - If two
polygons are similar, then the ratio of their
perimeters is equal to the ratios of their
corresponding side lengths.
Areas of Similar Polygons Theorem - If two polygons
are similar, then the ratio of their areas is equal
to the squares of the ratios of their
corresponding side lengths.
Angle-Angle (AA) Similarity Theorem - If two angles of
one triangle are congruent to two angles of
another triangle, then the two triangles are similar.
Side-Side-Side (SSS) Similarity Theorem - If the
corresponding side lengths of two triangles are
proportional, then the triangles are similar.
Side-Angle-Side (SAS) Similarity Theorem - If two sides
of one triangle are proportional to two sides of
another triangle and their included angles are
congruent, then the triangles are similar.
b
y
AB+BC+AC = AB = BC = AC
XY+YZ+XZ XY YZ XZ
z
x
c
a
b
y
2
b
y
If
∠A ∠X
x
c
a
b
2
( )( )( )
x
c
a
2
ΔABC area = AB = BC = AC
ΔXYZ area XY YZ XZ
z
z
and ∠B ∠Y
then
ΔABC ̴ ΔXYZ
If
y
AB = BC = AC
XY YZ XZ
x
c
a
b
z
y
then
ΔABC ̴ ΔXYZ
If
AB = AC
l and ∠A  ∠X
XY XZ

a
x
c
z
then
ΔABC ̴ ΔXYZ
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Similarity & Proportions
b
Triangle Proportionality Theorem - If a line parallel to one
side of a triangle intersects the other two sides,
then it divides the two sides proportionally.
Converse of the Triangle Proportionality Theorem - If a
line divides two sides of a triangle proportionally,
then it is parallel to the third side.


x
AX = CY
BX
BY
y
a
c
b
x
y
a
Corollary to the Triangle Proportionality Theorem
(Proportional Parts of Parallel Lines Corollary) If three parallel lines intersect two transversals,
then they divide the transversals proportionally.
Corollary to the Triangle Proportionality Theorem
(Congruent Parts of Parallel Lines Corollary) If three parallel lines cutoff congruent segments
on one transversal, then they cut off congruent
segments on every transversal.
Triangle Angle Bisector Theorem - An angle bisector of a
triangle divides the opposite side into two segment
whose lengths are proportional to the lengths of
a
the other two sides
c
a
e
f
b
c
c
AB = EF
BC
FG
g
e
If
f
AE||BF||CG & ABBC
g
EF FG
a
b
If
AX = CY
BX
BY
then,
XY || AC
then,
b
AX = AC
BX
BC
x
c
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Right Triangles & Trigonometry
Right Triangle Similarity Theorem - If an altitude is
drawn to the hypotenuse of a right triangle,
then the two triangles formed are similar to the
original triangle and each other.
Geometric Mean (Altitude) Theorem - In a right triangle,
the altitude from the right angle to the
hypotenuse divides the hypotenuse into two
segments. the lengths of the altitude is the
geometric mean of the lengths of the two
segments of the hypotenuse.
Geometric Mean (Leg) Theorem - In a right triangle, the
altitude from the right angle to the hypotenuse
divides the hypotenuse into two segments. The
length of each leg of the right triangle is the
geometric mean of the lengths of the
hypotenuse and the segment of the hypotenuse
that is adjacent to the leg.
b
d
ΔABC ~ ΔDBA ~ ΔDAC
a
b
c
x
d
h
a
b
c
x
d
c
a
a
h= √xy
y
a= √xc
and
y
b= √yc
c
b
b
Pythagorean Theorem - In a right triangle, the square
of the length of the hypotenuse is equal to the
sum of the squares of the lengths of the legs.
Converse of the Pythagorean Theorem - If the
square of the length of the longest side of a
triangle is equal to the sum of the squares of
the lengths of the other two sides, then the
triangle is a right triangle.
a
a
c
c
b
b
a
a
a2+b2=c2
c
b
If
a +b2=c2
then,
ΔABC is a right triangle
2
c
Pythagorean Inequality Theorem - For any ΔABC, where c is the length of the longest
side, the following statements are true:
If c2<a2+b2 , then ΔABC is acute.
If c2>a2+b2 , then ΔABC is obtuse.
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Right Triangles & Trigonometry
45°-45°-90° Triangle Theorem - In a 45°-45°-90° triangle, the
hypotenuse is √2 time as long as each leg.
s
s
45°
45°
s √2
30°-60°-90° Triangle Theorem - In a 30°-60°-90° triangle,
the hypotenuse is twice as long as the shorter leg,
and the longer leg is √3 times as long as the
shorter leg.
60°
s
2s
30°
s √3
Law of Sines Theorem - For any ΔABC that has lengths a, b, and c representing the
lengths of the sides opposite the angles with measures A, B, and C, then the
following is true: sinA sinB sinC
=
=
a
b
c
Law of Cosines Theorem - For any ΔABC that has lengths a, b, and c representing the
lengths of the sides opposite the angles with measures A, B, and C, then the
following is true:
a 2 =b 2 +c 2 -bc(cosA)
b 2 =a 2 +c 2 -ac(cosB)
c 2 =a 2 +b 2 -ab(cos C)
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Circles
b
Congruent Central Angles Theorem - In the same
circle, or in congruent circles, two minor arcs
are congruent if and only if their
corresponding central angles are congruent.
If
a
∠1
1
c
2
∠2
then,
AB CD
d
b
Arc Addition Postulate - The measure of an arc
formed by two adjacent arcs is the sum of
the measures of the two arcs.
Congruent Corresponding Chords Theorem - In
the same circle, or in congruent circles, two
minor arcs are congruent if and only if their
corresponding chords are congruent.
a
c
mABC = mAB + mBC
b
c
a
If
AB CD
then
AB CD
d
Perpendicular Chord Bisector Theorem - If a
diameter (or radius) of a circle is
perpendicular to a chord, then it bisects the
chord and its arc.
Converse of the Perpendicular Chord Bisector
Theorem - If one chord of a circle is a
perpendicular bisector of another chord,
then the first chord is a diameter.
b
x
AY BY
y
a
c
and
AX BX
b
x
a
If
XZ is a  bisector of AB,
then
XZ is a diameter of C
y
c
z
b
Equidistant Chords Theorem - In the same circle,
or in congruent circles, two chords are
congruent if and only if they are equidistant
from the center.
y
a
c
x
z
If
XY XZ
then
AB CD
d
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Circles
Measure of an Inscribed Angle Theorem - The
measure of an inscribed angle is one half
the measure of its intercepted arc.
b
a
m∠1 ½mAB
1
b
Inscribed Angles of a Circle Theorem - If two
inscribed angles of a circle intercept the
same arc (or congruent arcs), then the
angles are congruent.
a
2
∠1
& ∠2 both intercept AB
thus,
1
∠ 1  ∠ 2
c
Inscribed Right Angle Theorem - An inscribed
angle of a triangle intercepts a diameter or
semicircle if and only if the angle is a right
angle.
b
a
c
Inscribed Quadrilateral Theorem - If a
quadrilateral is inscribed in a circle, then its
opposite angles are supplementary.
d
∠A & ∠C are supplementary
and
∠B & ∠D are supplementary
a
b
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Circles
Tangent Line to a Circle Theorem - In a plane, a
line is tangent to a circle if and only if it is
perpendicular to a radius drawn to the point
of tangency.
If
l
External Tangent Congruence Theorem - If two
segments from the same exterior point
are tangent to a circle, then they are
congruent.
l  CT
c
then,
Line l is tangent to C
t
d
a
AD AB
c
b
c
Angles Inside the Circle Theorem - If two secants
or 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.
b
m∠1 = ½ (mAB+mCD)
1
a
Angles on the Circle Theorem - If a secant/chord
and a tangent intersect at the point of
tangency, then the measure of each angle
formed is one-half the measure of its
intercepted arc.
d
a
b
m∠1 = ½ mAT
and
1
m∠2 = ½ mABT
2
t
Angles Outside the Circle Theorem - If a tangent
and a secant, two tangents, or two secants
intersect outside a circle, then the measure
of the angle formed is one-half the
difference of the measures of the
intercepted arcs.
d
b
m∠A = ½ (mDE-mBC)
a
c
e
©Amazing Mathematics
Postulates, Theorems, & Corollaries
Circles
Segments of Chords Theorem - If two chords
intersect in a circle, then the products of
the lengths of the chord segments are
equal.
Segments of Secants Theorem - If two secants
intersect in the exterior of a circle, then
the product of the measures of one
secant segment and its external secant
segment is equal to the product of the
measures of the other secant and its
external secant segment.
Segments of Secants & Tangents Theorem - If a
tangent and a secant intersect in the
exterior of a circle, then the square of the
measure of the tangent is equal to the
product of the measures of the secant
and its external secant segment.
c
b
AE*EC = BE*ED
e
a
d
d
b
AB*AD = AC*AE
a
c
e
b
AB2 = AC*AD
a
c
d
©Amazing Mathematics