Download Postulates

A-B-C-D Theorem- Every Isometry preserves Angle
measure, Betweenness, Collinearity (lines), and
Distance.
Congruent Figures- Two figures F and G are congruent
figures ( F  G ) iff G is the image of F under a
reflection or composite of reflections.
Adjacent Angles- Two non-straight, non-zero angles are
adjacent iff a common side is interior to the angle
formed by the non-common sides.
Corresponding Angles- Any pair of angles in similar
locations with respect to the transversal and each line.
AIA = → Parallel Lines Theorem- If two lines are cut
by a transversal and for alternate interior angles of
equal measure, then the lines are parallel.
Corresponding Angles Postulate- If two coplanar lines
are cut by a transversal, corresponding angles have the
same measure, then the lines are parallel.
Angle Addition Property- If a ray is in the interior of an
angle, then the sum of the measure of the two angles is
the measure of the big angle.
Corresponding Parts in Congruent Figures (CPCF)
Theorem- If two figures are congruent, then any pair of
corresponding parts is congruent.
Angle Congruence Theorem- Two angles are congruent
iff they have the same measure.
Figure Reflection Theorem- If a figure is determined by
certain points, then its reflection image is the
corresponding figure determined by the reflection
images of those points.
Angle Symmetry Theorem- The line containing the
bisector of an angle is a symmetry line of the angle.
Betweenness Theorem- If B is between A and C, then
AB  BC  AC .
Bisector- A ray bisectors an angle iff the ray is in the
interior and the two newly formed angles are equal to
each other.
Center of a Regular Polygon Theorem- In any regular
polygon there is a point (its center) which is equidistant
from all its vertices.
Circle- A circle is the set of all point in a plane at a
certain distance (radius) from a certain point (center).
Commutative Property of Addition- a  b  b  a
Commutative Property of Multiplication-
ab  ba
Complementary- Angles are complementary iff
m1  m2  90 .
Flip-Flop Theorem- If F and F’ are points or figures and
r(F) = F’, then r(F’) = F.
Glide Reflection Theorem- Let
G  T r m be a glide
reflection, and let G(P) = P’. Then the midpoint of
segment PP’ is on m.
Isosceles Trapezoid Symmetry Theorem- the
perpendicular bisector of one base of an isosceles
trapezoid is the perpendicular bisector of the other base
and a symmetry line for the trapezoid.
Isosceles Trapezoid Theorem- In an isosceles trapezoid,
the non-base sides are equal in measure.
Isosceles Triangle Symmetry Theorem- The line
containing the bisector of the vertex angle of an
isosceles triangle is a symmetry line for the triangle.
Isosceles Triangle Theorem- If a triangle has two equal
sides, then the angles opposite them are equal.
Kite Diagonal Theorem- The symmetry diagonal of a
kite is the perpendicular bisector of the other diagonal
and bisects the two angles at the ends of the kite.
Kite Symmetry Theorem- The line containing the ends
of a kite is a symmetry line for the kite.
Linear Pair Theorem- If two angles form a linear pair,
then they are supplementary.
Linear Pair- Two non-straight, non-zero angles are a
linear pair iff they are adjacent and their non-common
sides are opposite rays.
Midpoint- The midpoint of a segment AB is the point M
on segment AB with AM  MB
Opposite Rays- ray AB and ray AC are opposite rays iff
A is between B and C.
Parallel Lines → AIA = Theorem- If two parallel lines
are cut by a transversal, then alternate interior angles
are equal in measure.
Parallel Lines and Slopes Theorem- Two non-vertical
lines are parallel iff they have the same slope.
Parallel Lines Postulate- If two lines are parallel and
are cut by a transversal, corresponding angles have the
same measure.
Parallel Lines- Two coplanar lines m and n are parallel
lines (m//n) iff they have no points in common, or they
are identical
Perpendicular Bisector Theorem- If a point is on the
perpendicular bisector of a segment, then it is
equidistant from the endpoints of the segment.
Perpendicular Lines and Slopes Theorem- Two
nonvertical lines are perpendicular iff the product of
their slopes is -1.
Perpendicular to Parallels Theorem- In a plane, if a
line is perpendicular to one of two parallel lines, then it
is perpendicular to the other.
Perpendicular- Two lines, segments, or rays are
perpendicular iff the lines containing them form a 90˚
angle
Polygon-Sum Theorem- The sum of the measures of the
angles of a convex polygon of n sides is
(n  2)  180 .
Properties of a Parallelogram- In any parallelogram:
each diagonal forms two congruent triangles, opposite
sides are congruent, and the diagonals intersect at their
midpoints.
Quadrilateral-Sum Theorem- The sum of the measures
of the angles of a convex quadrilateral is 360˚.
Rectangle Symmetry Theorem- Every rectangle has two
symmetry lines, the perpendicular bisectors of its sides.
Rhombus- Every Rhombus has two symmetry lines, its
diagonals.
Segment Congruence Theorem- Two segments are
congruent iff they have the same length.
Segment Symmetry Theorem- A segment has exactly
two symmetry lines: its perpendicular bisector, and the
line containing the segment.
Side-Switching Theorem- If one side of an angle is
reflected over the line containing the angle bisector, its
image is the other side of the angle.
Sufficient conditions for a Parallelogram Theorem- If,
in a quadrilateral: both pairs of opposite sides are
congruent, or both pairs of opposite angles are
congruent, or the diagonals bisect each other then the
quadrilateral is a parallelogram.
Supplementary- Angles are supplementary iff
m1  m2  180 .
Theorem (Isosceles Trapezoid)- In an isosceles
trapezoid, both pairs of base angles are equal in
measure.
Theorem (Isosceles)- In an isosceles triangle, the
bisector of the vertex angle, the perpendicular bisector
of the base, and the median to the base determine the
same line.
Theorem (Parallelogram)- If a quadrilateral is a
rhombus, then it is a parallelogram.
Triangle Congruence – SsA- If, in two triangles, two
sides and the angle opposite the longer of the two sides
in one are congruent respectively to two sides and the
angle opposite the corresponding side in the other, then
the triangles are congruent.
Theorem (Triangles)- If two triangles have two pairs of
angles congruent, then their third pair of angles is
congruent.
Triangle Congruence – SSS- If, in two triangles, three
sides of one are congruent to three sides of the other,
then the triangles are congruent.
Theorem (Parallel lines)- The distance between parallel
lines is constant.
Transitive Property of Congruence- If
and G  H , then F  H .
Transitive Property of Equality- If
then a  c .
F G
a  b and b  c ,
Transitive Property of Inequality- If
then a  c .
a  b and b  c ,
Transitivity of Parallelism Theorem- In a plane, l//m
and m//n, then l//n.
Trapezoid Angle Theorem- In a trapezoid, consecutive
angels between a pair of parallel sides are
supplementary.
Triangle Congruence – AAS- If, in two triangles, two
angles and a non-included side of one are congruent
respectively to two angles and the corresponding nonincluded side of the other, then the triangles are
congruent.
Triangle Congruence – ASA- If in two triangles, two
angles and the included side of the other, then the
triangles are congruent.
Triangle Congruence - HL Congruence Theorem- If, in
two right triangles, the hypotenuse and a leg of one are
congruent to the hypotenuse and a leg of the other, then
the two triangles are congruent.
Triangle Congruence – SAS- If, in two triangles, two
sides and the included angle of one are congruent to two
sides and the included angle of the other, then the
triangles are congruent.
Triangle Inequality – SAS- If two sides of a triangle are
congruent to two sides of a second triangle and the
measure of the included angle of the first triangle is less
then the measure of the included angle of the second,
then the third side of the first triangle is shorter than the
third side of the second.
Triangle Sum Theorem- The sum of the measures of the
angles of a triangle is 180˚.
Two Perpendiculars Theorem- If two coplanar lines l
and m are each perpendicular to the same line, then they
area parallel to each other.
Two Reflection Theorem for Rotation- The rotation
rm  rl , where m intersects l, “turns” figures twice the
non-obtuse angle between l and m, measured from l to
m, about the point of intersection of the two lines.
Two Reflection Theorem for Translations- If m//l, the
translation rm  rl slides figures two times the distance
between l and m, in the direction from l to m
perpendicular to those lines.
Unique Angle Assumption- Every angle has a unique
measure from 0˚ to 180˚.
Vertical Angles Theorem- If two angles are vertical
angles, then they have equal measures.
Vertical Angles- Two angles are vertical angles iff their
sides form two lines
The degree of a minor arc or semicircle AB of circle
O, written mAB, is the measure of central angle AOB.
A cylinder is the surface of a cylindric solid whose
base is a circle.
Prism-Cylinder Volume Formula
The degree of a major arc ACB of circle O, written
mACB, is 360˚- mAB.
A prism is the surface of a cylindric solid whose base is
a polygon.
Pyramid-Cone Volume Formula
(Right) Triangle Area Formula
Trapezoid Area Formula
A
1
hb
2
1
A  h(b1  b2 )
2
Parallelogram Area Formula
A  hb
Circumference of a Circle
C  d
1
Bh
3
Volume of a Sphere
4
V  r 3
3
A cone is the surface of a conic solid whose base is a
circle.
Surface Area of a Sphere
SA  4r 2
A pyramid is the surface of a conic solid whose base is
a polygon.
A sphere is the set of points in space at a fixed distance
(its radius) from a point (its center)
A line l is perpendicular to a plane X if and only if it
is perpendicular to every line in X through their
intersection.
Regular Pyramid-Right Cone Lateral Area Formula
A cylindric solid is the set of point between a region
and its translation image in space, including the region
and its image.
V 
Given a region (the base) and a point (the vertex) not in
the plane of a region, a conic solid is the set of points
between the vertex and all points of the base, together
with the vertex and base
A  r
Circle Area Formula
Point-Line-Plane Postulate
A. Given a line in a plane, there exists a point in
the plane not on the line. Given a plane in
space, there exists a point in space not on the
plane.
B. Every line is a set of points that can be put into
a one-to-one correspondence with the real
numbers, with any point on it corresponding to
0 and any other point corresponding to 1.
C. Through two points there is exactly one line.
D. On a number line, there is a unique distance
between two points.
E. If two points lie on a plane, the line containing
them lies in the plane.
F. Through three non-collinear points, there is
exactly one plane.
G. If two different place have a point in common,
then their intersection is a line
2
V  Bh
A plane section of a three-dimensional figure is the
intersection of the figure with a plane.
Two figures F and G in space are congruent figures if
and only if G is the image of F under a reflection or
composite of reflections.
A space figure F is reflection-symmetric if and only if
there is a plane M such that rM ( F )  F .
Right Prism-Cylinder Lateral Area Formula LA  ph
(p-perimeter of bases)
Prism-Cylinder Surface Area Formula
SA  LA  2B
LA 
1
lp (l-slant height, p-perimeter)
2
Distance Formula – The distance between two points
( x1 , y1 ) and ( x2 , y2 ) in the coordinate place is
( x 2  x1 ) 2  ( y 2  y1 ) 2
Equation for a Circle – The circle with center (h,k)
and radius r is the set of points (x,y) satisfying
( x  h) 2  ( y  k ) 2  r 2
Midpoint Formula – If a segment has endpoints (a,b)
and (c,d), its midpoint is (
ac bd
,
)
2
2
Midpoint Connector Theorem – The segment
connecting the midpoints of two sides of a triangle is
parallel to and half the length of the third side.
Size Change Distance Theorem – Under a size change
of magnitude k  0 , the distance between any two
points is k times the distance between their preimages.
Size Change Theorem – Under a size transformation:
A. Angle measures are preserved
B. Betweenness is preserved
C. Collinearity is preserved
D. Lines and their images are parallel
Pyramid Cone Surface Area Formula
SA  LA  B
Figure Size Change Theorem – If a figure is
determined by certain points, then its size change image
is the corresponding figure determined by the size
change images of those points.
A transformation is a similarity transformation if and
only if it is the composite of size changes and
reflections.
Law of the Contrapositive – A statement ( p  q )
and its contrapositive ( not  q  not  p ) are either
bother true or both false.
Two figures F and G are similar, written F~G, if and
only if there is a similarity transformation mapping one
onto the other.
Law of Ruling Out Possibilities – When p or q is true
and q is not true, then p is true.
Similar Figures Theorem – If two figures are similar,
then:
A. Corresponding angles are congruent
B. Corresponding lengths are proportional
Two statements p and q are contradictory if and only if
they cannot be true at the same time.
Fundamental Theorem of Similarity – If G~G` and k
is the ration of similitude, then
A. Perimeter(G`) = k*Perimeter(G)
B. Area(G`) = k2*Area(G)
C. Volume(G`) = k3*Volume(G)
The SSS Similarity Theorem – If the three sides of
one triangle are proportional to the three sides of a
second triangle, then the triangles are similar.
The AA Similarity Theorem – If two triangles have
two angles of one congruent to two angles of the other,
then the triangles are similar.
The SAS Similarity Theorem – If, in two triangles, the
ratios of two pairs of corresponding sides are equal and
the included angles are congruent, then the triangles are
similar.
Side-Splitting Theorem – If a line is parallel to a side
of a triangle and intersects the other two sides in distinct
points, it ‘splits’ these sides into proportional segments.
Side-Splitting Converse – If a line intersects
OP and
OX OY
OQ in distinct points X and Y so that
,

XP YQ
then XY // PQ .
Law of Detachment – If you have a statement or given
information p and a justification of the form p  q ,
you can conclude q.
A tangent to a circle is the line which intersects the
circle in exactly one point.
If a point is perpendicular to a radius of a circle at the
radius’ endpoint on the circle, then it is tangent to the
circle.
If a line is tangent to a circle, then it is perpendicular to
the radius drawn to the point of tangency.
Radius-Tangent Theorem – A line is tangent to a
circle if and only if it is perpendicular to a radius at the
radius’ endpoint on the circle.
Uniqueness of Parallels Theorem – Through a point
not on the line, there is exactly on parallel to the given
line.
In a glide reflection, the midpoint of the segment
connecting a point to its image lies on the glidereflection line.
Postulates of Euclid
1. Two points determine a line
2. A line segment can be extended
indefinitely along a line
3. A circle can be drawn with any center
and any radius
4. All right angles are congruent
5. If two lines are cut by a transversal,
and the interior angles on the same
side of the transversal add up to less
then 180, then the lines will intersect
on that side of the transversal.
An angle is an exterior angle of a polygon if and only
if it forms a linear pair with one of the angles of the
polygon.
Exterior Angle Theorem – In a triangle, the measure
of an exterior angle is equal to the sum of the measures
of the two nonadjacent interior angles.
Exterior Angles Inequality – In a triangle, the measure
of an exterior angle is greater then the measure of either
nonadjacent interior angle.
Unequal Sides Theorem – If two sides of a triangle are
not congruent, then the angles opposite them are not
congruent, and the larger angle is opposite the longer
side.
Unequal Angles Theorem – If two angles of a triangle
are not congruent, then the sides opposite them are not
congruent, and the longer side is opposite the larger
angle.
Exterior Angles of a Polygon Sum Theorem – In any
convex polygon, the sum of the measures of the exterior
angles, one at each vertex, is 360.
The geometric mean of the positive numbers a and b is
ab .
Right Triangle Altitude Theorem – In a right triangle
A. The altitude to the hypotenuse is the geometric
mean of the segments into which it divides the
hypotenuse; and
B. Each leg is the geometric mean of the
hypotenuse and the segment of the hypotenuse
adjacent to the leg.