This is a partial listing of the more
popular theorems, postulates and
properties
needed when working with Euclidean
proofs.
You need to have a thorough understanding of these items.
Your textbook (and your teacher) may want you to
remember these theorems with slightly different wording.
Be sure to follow the directions from your teacher. 
The
"I need to know, now!"
entries are highlighted in blue.
General:
Reflexive Property 
A quantity is congruent (equal) to itself. a = a 
Symmetric Property 
If a = b,
then b = a. 
Transitive Property 
If a = b and
b = c, then a = c. 
Addition Postulate 
If equal quantities are added to
equal quantities, the sums are equal. 
Subtraction Postulate

If equal quantities are subtracted
from equal quantities, the differences are equal. 
Multiplication
Postulate 
If equal quantities are multiplied
by equal quantities, the products are equal. (also Doubles of
equal quantities are equal.) 
Division Postulate 
If equal quantities are divided by
equal nonzero quantities, the quotients are equal. (also Halves of
equal quantities are equal.) 
Substitution
Postulate 
A quantity may be substituted for
its equal in any expression. 
Partition Postulate 
The whole is equal to the sum of its parts.
Also: Betweeness of Points: AB + BC = AC
Angle Addition Postulate: m<ABC + m<CBD = m<ABD 
Construction 
Two points determine a straight
line.

Construction 
From a given point on (or not on)
a line, one and only one perpendicular can be drawn to the line. 
Angles:
Right Angles 
All right angles are congruent.

Straight Angles 
All straight angles
are congruent.

Congruent Supplements 
Supplements of the same angle, or
congruent angles, are congruent. 
Congruent Complements 
Complements of the same angle, or
congruent angles, are congruent. 
Linear Pair 
If two angles form a
linear pair, they are supplementary.

Vertical Angles 
Vertical angles are congruent.

Triangle Sum 
The sum of the interior angles of a triangle is
180º.

Exterior Angle 
The measure of an exterior angle of a triangle is
equal to the sum of the measures of the two nonadjacent interior
angles.
The measure of an exterior angle of a triangle is greater than
either nonadjacent interior angle. 
Base Angle Theorem
(Isosceles Triangle) 
If two sides of a triangle are
congruent, the angles opposite these sides are congruent. 
Base Angle Converse
(Isosceles Triangle) 
If two angles of a triangle are
congruent, the sides opposite these angles are congruent. 
Triangles:
SideSideSide (SSS) Congruence 
If three sides of one triangle are congruent to
three sides of another triangle, then the triangles are
congruent. 
SideAngleSide (SAS) Congruence 
If two sides and the included angle of one triangle
are congruent to the corresponding parts of another triangle, the
triangles are congruent. 
AngleSideAngle (ASA) Congruence 
If two angles and the included side of one triangle
are congruent to the corresponding parts of another triangle, the
triangles are congruent. 
AngleAngleSide (AAS) Congruence 
If two angles and the nonincluded side
of one triangle are congruent to the corresponding parts of another
triangle, the triangles are congruent. 
HypotenuseLeg (HL) Congruence (right triangle) 
If the hypotenuse and leg of one right
triangle are congruent to the corresponding parts of another right
triangle, the two right triangles are congruent. 
CPCTC 
Corresponding parts
of congruent triangles are congruent. 
AngleAngle (AA) Similarity 
If two angles of one triangle are congruent to two
angles of another triangle, the triangles are
similar. 
SSS for Similarity 
If the three sets of
corresponding sides of two triangles are in proportion, the
triangles are similar. 
SAS for Similarity 
If an angle of one
triangle is congruent to the corresponding angle of another triangle
and the lengths of the sides including these angles are in
proportion, the triangles are similar. 
Side Proportionality 
If two triangles are
similar, the
corresponding sides are in proportion. 
Midsegment Theorem
(also called midline) 
The segment connecting the midpoints of
two sides of a triangle is parallel
to the third side and is half as
long. 
Sum of Two Sides 
The sum of the
lengths of any two sides of a triangle must be greater than the
third side

Longest Side 
In a triangle, the longest side is
across from the largest angle.
In a triangle, the largest angle is across from the longest side. 
Altitude Rule 
The altitude
to the hypotenuse of a right triangle is the mean proportional
between the segments into which it divides the hypotenuse. 
Leg Rule 
Each leg
of a right triangle is the mean proportional between the hypotenuse
and the projection of the leg on the hypotenuse. 
Parallels:
Corresponding Angles

If two parallel lines are cut by a transversal, then
the pairs of corresponding angles are congruent. 
Corresponding Angles Converse

If two lines are cut by a transversal and the
corresponding angles are congruent, the lines are
parallel. 
Alternate Interior Angles

If two
parallel lines
are cut by a transversal, then the alternate
interior angles are congruent. 
Alternate Exterior Angles 
If two
parallel lines are cut by a
transversal, then the alternate exterior angles
are congruent. 
Interiors on Same Side

If two
parallel lines
are cut by a transversal, the interior angles on
the same side of the transversal are
supplementary. 
Alternate Interior Angles
Converse 
If two lines
are cut by a transversal and the alternate
interior angles are congruent, the lines are
parallel. 
Alternate Exterior Angles
Converse 
If two lines
are cut by a transversal and the alternate
exterior angles are congruent, the lines are
parallel. 
Interiors on Same
Side Converse 
If two lines are cut by a
transversal and the interior angles on the same
side of the transversal are supplementary, the
lines are parallel. 

Quadrilaterals:
Parallelograms

About Sides

* If a quadrilateral
is a parallelogram, the opposite
sides are parallel.
* If a
quadrilateral is a parallelogram, the opposite
sides are congruent. 
About Angles 
* If a quadrilateral
is a parallelogram, the opposite
angles are congruent.
* If a
quadrilateral is a parallelogram, the
consecutive angles are supplementary. 
About Diagonals 
* If a quadrilateral
is a parallelogram, the diagonals
bisect each other.
* If a
quadrilateral is a parallelogram, the diagonals
form two congruent triangles. 
Parallelogram Converses

About Sides

* If both pairs of
opposite sides of a quadrilateral
are parallel, the quadrilateral is a parallelogram.
* If both pairs of
opposite sides of a quadrilateral
are congruent, the quadrilateral is a
parallelogram. 
About Angles 
* If both pairs of
opposite angles of a quadrilateral
are congruent, the quadrilateral is a
parallelogram.
* If the
consecutive angles of a quadrilateral are
supplementary, the quadrilateral is a parallelogram. 
About Diagonals

* If the diagonals of
a quadrilateral bisect each
other, the quadrilateral is a
parallelogram.
* If the diagonals
of a quadrilateral form two
congruent triangles, the quadrilateral is a
parallelogram. 
Parallelogram 
If
one pair of sides of a quadrilateral is BOTH parallel and
congruent, the quadrilateral is a parallelogram. 
Rectangle 
If a
parallelogram has one right angle it is a rectangle 
A
parallelogram is a rectangle if and only if its diagonals are
congruent. 
A
rectangle is a parallelogram with four right angles. 
Rhombus 
A
rhombus is a parallelogram with four congruent sides. 
If a parallelogram has two consecutive sides congruent, it is a
rhombus. 
A
parallelogram is a rhombus if and only if each diagonal bisects
a pair of opposite angles. 
A
parallelogram is a rhombus if and only if the diagonals are
perpendicular. 
Square 
A
square is a parallelogram with four congruent sides and four
right angles. 
A
quadrilateral is a square if and only if it is a rhombus and a
rectangle. 
Trapezoid 
A
trapezoid is a quadrilateral with exactly one pair of parallel
sides. 
Isosceles Trapezoid 
An
isosceles trapezoid is a trapezoid with congruent legs. 
A
trapezoid is isosceles if and only if the base angles are
congruent 
A
trapezoid is isosceles if and only if the diagonals are
congruent 
If a
trapezoid is isosceles, the opposite angles are supplementary. 
Circles:
Radius 
In a circle, a radius
perpendicular to a chord bisects the chord and the arc. 
In a circle, a radius that bisects
a chord is perpendicular to the chord. 
In a circle, the perpendicular bisector of a chord
passes through the center of the circle.

If a line is tangent to a circle,
it is perpendicular to the radius drawn to the point of tangency. 
Chords 
In a circle, or congruent circles, congruent
chords are equidistant from the center. (and converse)

In a circle, or congruent circles,
congruent chords have congruent arcs. (and converse0 
In a circle, parallel chords
intercept congruent arcs 
In the same circle, or congruent
circles, congruent central angles have congruent chords (and
converse) 
Tangents 
Tangent segments to a circle from
the same external point are congruent 
Arcs 
In the same circle, or congruent
circles, congruent central angles have congruent arcs. (and
converse) 
Angles 
An angle inscribed in a
semicircle is a right angle. 
In a circle, inscribed angles that intercept the
same arc are congruent. 
The opposite angles in a cyclic
quadrilateral are supplementary 
In a circle, or congruent circles,
congruent central angles have congruent arcs. 
