# Math Theorems

## Terms

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- Commutative Property of Addition
- a+b=b+a
- Commutative Property of Multiplication
- ab=ba
- Associative Property of Addition
- (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
- Addition Property
- 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
- Segment Addition Postulate
- If point B is on line AC and between points A and C, then segment AB + segment BC = line AC.
- Angle Addition Postulate
- 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.
- Quadrilateral Sum Theorem
- 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.
- Arc Addition Postulate
- 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.
- Cylic Quadrilateral Theorem
- 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.