Segment Addition Postulate
If three points, A, B, and C are colinear and B is between A and C, then AB+BC=AC
Addition Property (Equality)
If a=b, then a+c = a+b
Subtraction Property (Equality)
If a=b, then a-c = b-c
Multiplication Property (Equality)
If a=b, then ac = bc
Division Property (Equality)
If a=b and c does not equal 0, then a/c = b/c
Reflexive Property (Equality)
a=a
is used the most
Symmetric Property (Equality)
If a=b, then b=a
Transitive Property (Equality)
If a=b and b=c, then a=c
Substitution Property (Equality)
If a=b, then b can replace a in any expression
The Distributive Property (Equality)
a(b+c)= ab+ac
Reflexive Property (Congruence)
Line AB is congruent to line AB
Angle A is congruent to angle A
Symmetric Property (Congruence)
If line AB is congruent to line CD, then line CD is congruent to line AB
If angle A is congruent to angle B, then angle B is congruent to angle A
Transitive Property (Congruence)
If line AB is congruent to line CD and line CD is congruent to line EF, then line AB is congruent to line EF
If angle A is congruent to angle B and angle B is congruent to angle C, then angle A is congruent to angle C
Vertical Angles Theorem
Vertical angles are congruent
Congruent Supplements Theorem
If two angles are supplements of the same angle, then the two angles are congruent
Congruence Complements Theorem
If two angles are complementary of the same angle, then the two angles are congruent
Theorem:Right Angles
All right angles are congruent
Theorem
If two angles are congruent and supplementary, then each is a right angle
Corresponding Angles Postulate
If two parallel lines are cut by a transversal, corresponding angles are congruent
Alternate Interior Angle Theorem
If a transversal intersects two parallel lines, then alternate interior angles are congruent
Same Side Interior Angle Theorem
If a transversal intersects two parallel lines, then same side interior angles are supplementary
Converse Of The Corresponding Angles Postulate
If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel
ConverseOf The Alternate Interior Angles Theorem
If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel
Converse Of The Same Side Interior Angles Theorem
If two lines and a transversal form same side interior angles that are supplementary, then the two lines are parallel
Theorem:Parallel Lines
If two lines are parallel to the same line, then they are parallel to each other
Theorem
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other
Triangle Angle Sum Theorem
Measurements of interior angles of a triangle add up to 180 degrees
Triangle Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles
Triangle Third Angle Theorem
If two angles of one triangle are congruent to two angles of another triangle, then their third angles are congruent
Side-Side-Side Postulate (SSS Postulate)
If three sides of one triangle are congruent to three sides of another triangle then the triangles are congruent
Side-Angle-Side Postulate (SAS 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 two triangles are congruent
Angle-Side-Angle Postulate (ASA Postulate)
If two angles and an included side of one triangle are congruent to two angles and a included side of another triangle then the two triangles are congruent
Angle-Angle-Side Theorem (AAS Theorem)
If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle then the two triangles are congruent
The Isosceles Triangle Theorem
If two sides of a triangle are congruent then the angles opposite those angles are congruent
Converse Of The Isosceles Triangle Theorem
If two angles of a triangle are congruent then the side a opposite the angles are congruent
Theorem
The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base
RHL Theorem
If a hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle then the two triangles are congruent
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equal distance to the end points of that segment.
Converse of the Perpendicular Bisector Theorem
If a point is equal distance to the end points of a segment, then it is on the perpendicular bisector of that segment.
Angle Bisector Theorem
If a point is on the angle bisector of an angle then it is equal distance to the sides of the angle.
Converse of the Angle Bisector Theorem
If a point is equal distance to the sis of an angle the. It is I the angle bisector of than angle.
Theorem
The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant to the vertices of the triangle.
Theorem
The angle bisectors of a triangle are concurrent at a point equidistant to the side of the triangle.
Corollary to the triangle exterior angle theorem
The measure of an exterior angle of a triangle is greater than the measure of each remote interior angle
Theorem
If two sides of a triangle are not congruent, then the larger angle lies opposite the larger side
Theorem
If two angles of a triangle are not congruent, then the longer side lies opposite the opposite the larger angle.
Theorem
The sum of the lengths of any two sides of a triangle must be greater than the length of the third (the largest) side.
Theorem
In a parallelogram opposite sides are congruent.
Theorem
Consecutive angles in a parallelogram are supplementary.
Theorem
Opposite angles in a parallelogram are congruent
Theorem
In a parallelogram the diagonals bisect each other
Theorem
If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal
Theorem
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem
In a quadrilateral, if both pair of opposite angles are congruent, then the quadrilateral is a parallelogram.
Theorem
If the diagonals of a quadrilateral bisect each other then the quadrilateral is a parallelogram.
Theorem
If one pair of sides of a quadrilateral are BOTH parallel and congruent then the quadrilateral is a parallelogram.
Theorem
In a rhombus the diagonals bisect the angles.
Theorem
In a rhombus the diagonals are perpendicular.
Theorem
In a rectangle the diagonals are congruent.
Theorem
If one diagonal of a parallelogram bisects two of the angles, then the parallelogram is a rhombus.
Theorem
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
Theorem
If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
Theorem
In an isosceles trapezoid the base angles are congruent.
Theorem
The diagonals of an isosceles trapezoid are congruent
Theorem
The diagonals of a kite are perpendicular.
Trapezoid Midsegment Theorem
The Midsegment if a trapezoid is parallel to both bases and is half the sum (average) of the bases.
Pythagorean theorem
A squared + B squared = C squared
Angle Angle Similarity (AA~)
If two angles of one triangle are congruent to two angles of another triangle, the two triangles are similar
SAS~ Theorem (Side Angle Side Similarity Theorem)
If an angle of one triangle is congruent to an angle of another triangle, and the sides including the angle are proportional, the two triangles are similar
SSS~ Theorem (Side Side Side Similarity Theorem)
If the corresponding sides of two triangles are proportional, then the triangles are similar
Theorem
The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to each other and to the original triangle
Geometric Mean
The geometric mean of a and b is the positive such that a/x = x/b
Corollary
The length of the altitude drawn to the hypotenuse of a right triangle is the geometric mean between the lengths of the segments of the hypotenuse
Corollary
The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse so that the length of each leg of the triangle is the geometric mean between the length of the adjacent hypotenuse segment and the length of the hypotenuse
Side Splitter Theorem
If a line is parallel to one side of a triangle, and intersects the other two sides then it divides those sides proportionally.
Corollary
If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional
Triangle Angle Bisector Theorem
If a line bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle
Theorem
If the similarity ratio of two similar figures is a/b then the perimeter ratio is a/b and the area ratio is a squared/ b squared
Theorem
The tangent to a circle is perpendicular to a radius