School Geometry Theorems

Theorem on the congruency of 2 right angles

If two angles are right angles, then they are congruent

Theorem on the congruency of straight angles to eachother

If two angles are straight angles, then they are congruent

Theorem on angles being supplementary to the same angle

If angles are supplementary to the same angle then they are congruent

Theorem on an angle that is complementary to other angles

If angles are complementary to the same angle, then they are congruent

Theorem

A mathematical statement that can be proved

Distributive POE

If a ( b +, - c ), then ab + , - ac

Reflexive POE

a = a

Symmetric POE

If a = b, then b = a

Transitive POE

If a = b and b = c, then a = c

Substitution POE

If a = b, then a can be substituted for b

Division POE

If a = b, a/c =b/c

Multiplication POE

If a = b, then ac = bc

Addition/Subtraction Postulate

If a=b, a+c = b+c

Contrapositive Statement

~q --> ~p if not q then not p

Inverse Statement

~p --> ~q (if not p then not q)

Biconditional Statement

Both p --> q and q --> p are true

Converse Statement

q --> p (if q then p)

Conditional Statement

p --> q (if p then q)

Addition Property (Theorem on a segment being added to two congruent segments)

If a segment is added to two congruent segments, then the sums are congruent

Addition property

(Theorem on congruent segments being added to other segments that are congruent to each other)

If congruent segments are added to congruent segments that the sums are congruent

Addition property

Theorem on congruent angles being added to other congruent angles

If congruent angles are added to congruent angles, then the sums are congruent

Addition property

Theorem on angle being added to congruent angles

If an angle is added to congruent angles then the sums are congruent

Subtraction property

Theorem on a segment (or angle) being subtracted from a congruent segment (or angle)

If a segment (or angle) is subtracted from congruent segments (or angles), then the differences are congruent

Subtraction Property

Theorem on congruent segments (or angles) being subtracted by other congruent segments (or angles)

If congruent segments (or angles) are subtracted from congruent segments (or angles) then the differences are congruent.

Perpendicularity converse definition

If an angle is right then it has been formed by perpendicular lines or segments

Perpendicularity

If two lines/segments are perpendicular they form right angles

Angle Bisector

1. If a ray bisects an angle, then it divides the angle into two congruent angles

2. If an angle is divided into 2 congruent angles, then it has been bisected

Bisector of a segment

1. If a line (or ray or segment) bisects a segment, then it divides the segment into 2 congruent segments

2. If a segment is divided into 2 congruent segments by a line (or ray or segment) then it bisects the segment

Congruent Angles

1. If the measures of 2 angles equal, then they are congruent

2. If two angles are congruent, then they have equal measures.

Congruent Segments

1.If 2 segments have equal lengths, then they are congruent.

2. If 2 segments are congruent, then they have equal lengths.

Multiplication Property

If segments (or angles) are congruent, then their like multiples are congruent

Division Property

If segments (or angles) are congruent, then their like divisions are congruent

Reflexive Property

If there is a segment or angle, then it is congruent to itself

SSS Postulate

SIde Side Side

SAS Postulate

Side Angle Side

ASA

Angle Side Angle

Postulate on constructing a line with two points

Two points determine a line

Corresponding Parts of Congruent Triangles are Congruent

CPCTC

Theorem on the congruency of radii in a circle

The radii of a circle are congruent

Theorem on the congruency of vertical angles

Vertical angles are congruent

Postulate on rigid motion

Rotations are a rigid motion

Theorem of congruent angles being supplementary to other angles

If angles are supplementary to congruent angles, then they are congruent

Theorem of congruent angles being complementary to other angles

If angles are complementary to congruent angles, then they are congruent

Segment Addition Postulate

AB + BC = AC

Angle addition Postulate

m< BAC + m<CAD = m<BAD

Congruent Segments

AB is congruent to CD <=> AB = CD

Congruent Angles

<P is congruent to <Q <=> m<P = m<Q

ITT (Isoceles triangle theorem)

If two sides of a triangle are congruent, then the angles opposite the sides are congruent

ITTC (Isosceles triangle Theorem Converse)

If two angles of a triangle are congruent, then the sides opposite the angles are congruent.

Inverse of ITT

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

Inverse of ITTC

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

HL postulate

If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other triangle, the two right triangles are congruent.

Midpoint formula

Mx=(x1+x2)/2; My(y1+y2)/2

Right Angle Theorem (RAT)

If tow angles are both supplementary and congruent, then they are right angles

EDTC

If a point is on the perpendicular bisector of the segment, then it is equadistant from the endpoints of the segment

EDT

If two points are each equidistant from the endpoints of a segment, then the points determine the perpendicular bisector of the segment.

Way to prove parallel lines with alternate interior angles

alternate interior angles congruent

Way to prove parallel lines with alternate exterior angles

alternate exterior angles congruent

Way to prove parallel lines with corresponding angles

corresponding angles congruent

Way to prove parallel lines with same side interior angles

same side interior angles supplementary

Way to prove parallel lines with same side exterior angles

same side exterior angles supplementary

What is the sum of all three angles in a triangle

The sum of measures of a triangle in a triangle is 180

Theorem on the meusure of the exterior angle with correlation to two remote interior angles

The measure of an exterior angle is equal to the sum of the measures of the two remote interior angles

Theorem of a segment of a triangle with endpoints being on the midpoints

A segment joining the midpoint of a triangle is parallel to the third side, and its length is one-half the length of the third side. (Midline Theorem)

No choice Theorem

If two angles of one triangle are congruent of two angles of a second triangle, then the third angles are congruent

AAS Theorem

If there exists a correspondence between the verticies of two triangles such that two angles and the NON - INCLUDED side of one are congruent to the corresponding parts of the other then the triangles are congruent

The sum Si of the measure of the angles of a polygon with n sides is given by the formula:

180(n-2)

If one exterior angle is taked at each vertex, the sum Se of the measures of the exterior angles of a polygon is given by the formula

360

The number d of diaganols that can be drawn in a polygon of n sides is given by the formula:

n(n-3)/2

Regular Polygon

A polygon both equilateral and equilangular

The measure of E of each exterior angle of equilangular polygon of n sides is given by formula

360/n

MEPT

In a proportion, the product of the means is equal to the product of the extremes; a/b=c/d then ad = bc

MERT

If the product of a pair of non - zero numbers is equal to the product of another pair of non - zero numbers, then either pair of numbers may be extremes, and the others the means of the proportion. If pq = rs then p/r=s/q, p/s = r/q, r/p = q/s

The theorem on the relation of the perimeters of polygons and there corresponding sides

The ratio of the perimiters of two similar polygons is equal to the ratio of any corresphonding sides corresponding

AAA Postulate

If there exists a correspondence between the vertices of two triangles such that the three angles of one triangle are congruent to the corresponding angles of the other triangle, then the triangles are similar

AA Theorem

If there exists a correspondence between the verticies of two triangles such that TWO of the angles of one triangle are congruent to the corresponding angles of the other then the triagles are silmilar.

SSS~ Theorem

If there exists a correspondence between the vertices of two triangles such that the ratio of the measures of corresponding sides are equal then the triangles are similar

SAS~ Theorem

If there exists a correspondence between the vertices of two triangles that the ratios of the measures of two pairs of corresponding sides are equal and the included angles are congruent then the triangles are similar.

CSSTP

Definitions of Similar Triangles:

1. Corresponding sides of similar triangles are proportional

2. Corresponding sides of similar triangles are congruent

Side - Splitter Theorem

If a line is parallel to one side of a triangle and intercepts the other two sides, it divides those two sides proportionally.

Angle Bisector Theorem

If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides

Theorem on the proportions of three or more parallel lines with a transversal

If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversal proportionally

Definition of Dilation

A transformation in which all points of a figure are enlarged or reduced by scale factor K, a center or rotation C such that

CP’/CP=|K|

Enlargement

If |K| >1 for a dilation

Reduction

If |K| <1 for a dilation

Notation for a dilation

D(C,K)

If P (x,y) is the preimage of a point, then its image after a dilation centered at the origin with scale factor K is the point:

P’(kx,ky)

Chord of a Circle

Line segment joining 2 points on a circle

Arc of a circle

An arc is made up of 2 points on a circle and all the points of the circle needed to connect them on a single path

Sector of a Circle

Portion of a circle

Inscribed angle of a circle

An angle formed by 2 chords

Theorem on the similarity of the triangles within a right triangle formed by a hypoteneuse

If an altitude is drawn to the hypotenuse of a right triangle, then all three triangles are congruent

Theorem on the proportions of the sides of the triangles within a right triangle formed by a hypotenuse

If an altitude is drawn to the hypotenuse of a right triangle, then there exists a mean proportional between two other sides of the triangle for each of the three sets of triangles

AHT

If an altitude is drawn to the hypotenuse of a right triangle, then either leg of a given triangle set is the mean proportional between the hypotenuse of the given triangle and the projection of that Leg

Pythagoream Theorem

a²+b²=c²

Converse of pythagoream theorem

If a²+b²=c², then the triangle is a right triangle

If P = (x₁,y₁) and Q = (x₂,y₂) are any two points, then the distance between them can be found with the formula

((x₁-x₂)²+(y₁-y₂)²)^1/2

Principle of the reduced triangle

Identify side lengths as a multiple of a pythagorean triple

Sine of A

opposite / hypotenuse

Cosine of A

adjacent / hypoteneuse