# Math Theorems

## Terms

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a+b=b+a
Commutative Property of Multiplication
ab=ba
(a+b)+c= a+(b+c)
Associative Property of Multiplication
(ab)c=a(bc)
Distributive Property
a(b+c)= ab+ac
Reflexive Property
a=a
Transitive Property
If a=b and b=c then a=c
Symmetric Property
If a=b then b=a
If a=b, then a+c=b+c. Also, if a=b and c=d, then ac=bd.
Division Property
If a=b, then a/c=b/c, provided C is not 0. Also, If a=b and c=d, then a/c=b/d, if C is not 0 and D is not 0.
Square root Property
If a squared = b, then a=+- the square root of b.
Zero Product
if ab=0, then a=0 or b=0, or both a and b = zero.
Line Postulate
Exactly one line can be constructed through 2 points
Line Intersection Postulate
The intersection of 2 distict lines is at exactly one point
Segment Duplication Postulate
Exactly one segmant can be constructed congruent to another segment.
Angle Duplication Postulate
Exactly one angle can be constructed congruent to another angle
Midpoint Postulate
Exactly one midpoint can be constructed on any line
Angle Bisector Postulate
Exactly one angle bisector bacn we constructed on any angle.
Parallel Postulate
Through one point not on a given line, exactly one parallel line to the given line can be constructed.
Perpendicular Postulate
THrough a point not on a line, exactly one line perpendicular to the line can be constructed
If point B is on line AC and between points A and C, then segment AB + segment BC = line AC.
If point D lies in the interior of angle ABC, then the measure of angle ABD + the measure of angle DBC = the measure of angle ABC.
Linear Pair Postulate
If 2 angles are a linear pair, they are supplementary
Cooresponding Angles Postulate
If 2 parallel lines are cut by a transveral, then the cooresponding andlges are congruent.
Converse to the Cooresponding Angles Postulate
If 2 lines are cut by a transveral forming cooresponding angles, then the lines are parallel.
SSS Congruence Postulate
If the 3 sides of a triangle are congruent to the 3 sides of another triangle, then the 2 triangles are congruent.
SAS Congruence Postulate
If 2 sides and their included angle in one triangle are congruent to 2 sides and the included angle of another triangle, the triangles are congruent.
ASA Congruence Postulate
If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle, then the triangles are congruent.
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
Converse of the Alternate Interior Angles Theorem
If alternate interior angles are congruent, then the twi lines cut by the transveral are parallel.
Triangle Sum Theorem
The sum of the measures of the anlges in a triangle is 180 degrees.
Third Angle Theorem
If two angles of one triangle are congruent to two angles of a second triangle, then the third pair of angles are congruent.
Congruent and Supplementary Theorem
If two angles are both congruent and supplementary, then each is a right angle.
Supplements of Congruent Angles Theorem
Supplements of congruent angles are congruent.
Right Angles Are Congruent Theorem
All right angles are congruent.
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.
Converse of the Alternate Exterior Angles Theorem
If two lines are cut by a transveral forming congruent alternate Exterior angles, then the lines are parallel.
Interior Supplements Theorem
If two lines are cut by a transveral, then the interior angles on the same side of the transveral are supplementary.
Converse of the Interior Supplements Theorem
If two linens are cut by a transversal forming interior angles on the same side of the transversal that are supplementary, then the lines are parallel.
Parallel Transitivity Theorem
If two lines in the same plane are parallel to the third line, they are parallel to each other.
Perpendicular to Parallel Theorem
If two lines in the same plane are perpendicular to a third line, then they are parallel to each other.
SAA Congruence Theorem
If 2 angles and a non included side of a triangle are congruent to the corresponding angles and non included side of another triangle, then the triangles are congruent.
Angle Bisector Theorem
Any point on the bisector of an angleis equidistant from the sides of the angle.
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equally distant from the endpoints of the segment.
Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it is the perpendiculr bisector of the segment.
Isosceles Triangle Theorem
If a triangle is isosceles, then the base angles are congruent.
Converse of the Isosceles Theorem
If two angles of a triangle are congruent, then the triangle is isosceles
Converse of the Angle Bisector Theorem
If a point is equally distant from the sides of an angle, then it is on the bisector of the angle.
Perpendicular Bisector Concurrency Theorem
The three perpendiculare bisectors of the sides of a triangle are concurrent.
Angle Bisector Concurrency Theorem
The three angle bisectors of the sides of a triangle are concurrent.
Triangle Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remore interior angles.
THe sum of the measures of the four angles of a quadrilateral is 360 degrees.
Medians to the Congruent Sides Theorem
In an isosceles triangle, the medians to the congruent sides are congruent.
Angle Bisectors to the Congruent Sides Theorem
In an isosceles triangle, the altitudes to the congruent sides are congruent.
Altitudes to the Congruent Sides Theorem
In an isosceles triangle, the altitudes to the congruent sides are congruent.
Isosceles Triangle Vertex Angle Theorem
The bisector of the vertex angle of an isosceles triangle is also the median and altitude to the base.
Parallelogram Diangonal Lemma
A diagonal of a parallelogram divides the parallelogram into two congruent triangles.
Opposite Sides Theorem
The opposite sides of a parallelogram are congruent
Opposie Angles Theorem
The opposite angles of a parallelogram are congruent
Converse of the Opposite Sides Theorem
If a figure is a parallelogram, then its opposite sides are congruent.
Converse of the Opposite Angles Theorem
If the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Oppiste Sides Parallel and Congruent Theorem
If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.
Rhombus Angles Theorem
Each diagonal of a rhombus bisects two opposite angles.
Parallelogran Consecutive Angles Theorem
The consecutive angles of a parallelogram are supplementary.
Four Congruent Sides Rhombus Theorem
If a quadrilateral has four congruent sides, then it is a rhombus.
Four Congruent Angles Rectangle Theorem
If a quadrilateral has four congruent angles, it is a rectangle
Rectangle Diagonals Theorem
The diangonals of a rectangle are congruent
Converse of the Rectangle Diangonals Theorem
If the diangonals of a prallelogram are congruent, then the parallelogram is a rectangle.
Isosceles Trapezoid Theorem
The base angles of an isosceles trapezoid are congruent.
Isosceles Trapezoid Diagonals Theorem
The diagonals of an isosceles trapezoid are congruent.
Converse of the Rhombus Angles Theorem
If a diagonal of a parallelogram bisects two opposite angles, then the parallelogram is a rhombus.
Double Edged Straight Edge Theorem
IF two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a rhombus.
Tangent Theorem
A tangent is perpendicular to the radius drwan to the point of tangency.
Theorem
No triangle has two right angles.
Perpendicular Bisector of a Chord Theorem
The perpendicular bisector of a chord passes through the center of the circle.
If point B is one arc AC and between points A and C, then the measure of arc AB + the measure of arc BC = the measure of arc AC.
Inscribed Angle Theorem
The measure of an angle in a circle is half the measure of the central angle.
Inscribed Angles Intersecting Arcs Theorem
Inscribed angles that intercept the same or congruent arcs are congruent.
The opposite angles of an inscribed quadrilateral are supplementary.
Parallel Secants Congruent Arc Theorem
Parallel lines intercept cingruent arcs on a circle.
Parallelogram Inscribed in a Circle Theorem
If a parallelogram is inscribed within a circle, then the parallelogram is a rectangle.
Tangent Segments Theorem
Tangent segments from a point to a cirlce are congruent.
Intersecting Chords Theorem
The measure of an angle formed by two intersecting chords is half the sum of the measure of the two intercepted arcs.
Reflexive Property of Similarity
Any figure is similar to itself
Symmetric Property of Similarity
If Figure A is similar to Figure B, then Figure B is similar to Figure A
Transitive Property of Similarity
If Figure A is similar to Figure B and Figure B is similar to Figure C, then Figure A is similar to Figure C.
AA Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
SAS Similarity Theorem
If two sides of a triangle are proportional to two sides of another triangle and the included angles are congruent then the two triangles are similar.
SSS Similarity Theorem
If the 3 sides of one triangle are proportional to the 3 sides of another triangle, then the 2 triangles are similar.
Intersecting Secants Theorem
If, then
Corresponding Altitudes Theorem
If two triangles are similar, then corresponding altitudes are proportional to the corresponding sides.
Corresponding Medians Theorem
If two triangels are similar, then corresponding medians are proportional to the corresponding sides.
Corresponding Angle Bisector Theorem
If two triangles are similar, then corresponding angle bisectors are proportional to the corresponding sides.
Parallel Proportionality Theorem
If a line passes through two sides of a triangle parallel to the third side, then it dived the two sides proportionally.
Converse of the Parallel Proportionally Theorem
If a line passes through two sides of a triangle dividing them proportionally, then it is parallel to the third side.
Three Similar Right Triangles Theorem
If you drop an altitude from the vertex of a right angle to its hypotenuse, then it divides the right triangle into two right triangles that are similar to each other and to the other right triangle.
Altitude to Hypotenuse Theorem
The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments on the hypotenuse.
Pythagorean Theorem
In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. If a and b are the lengths of the legs, and c is the length of the hypotenuse then a squared + b squared = c squared.
Converse of the Pythagorean Theorem
If the lengths of the three sides of a triangle satisfy the Pythagorean equation, then the triangle is a right triangle.
Hypotenuse Leg Theorem
If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse of one leg and another right triangle, then the two right triangles are congruent.
Coordinate Midpoint Property
IF x,y and x2,y2 are the coordinates of the endpoints of a segment, then the coordinates of the midpoint are x1+x2/2,y1+y2/2
Parallel Slope Property
In a coordinate plane, two distinct lines are parallel if and only if their slopes are equal.
Perpendicular Slope Property
In a coordinate plane, two nonvertical lines are perpendicular if and only if their slopes are negatvie reciprocals of each other.
Distance Formula
The distance between points A(x1,y1) and B (x2,y2) is given by ABsquared= (x2-x1)=(y2-y1) or AB= the square root of (x2-x1) + (y2-y1)
Square Diagonals Theorem
The diagonals of a square are congruent and are perpendicular bisectors of each other.

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