## Geometry Theorems and Postulates

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Addition Property of Equality For real numbers a,b, and c, if a=b, then a+c=b+c
Additive Indenity The sum of any real number and zero is that same real number. In other words, for any real number a, a+0=a
Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then alternate interior angles are congruent
Alternate Exterior Angles Theorem If a transversal intersect two parallel lines, then alternate interior angles are congruent.
Angle Addition Postulate Thye measure of an angle created by two adjacent angles can be found by adding the measures of the two adjacent angles.
Angle-Angle(AA) Similarity Postulate If two corresponding angles of two or more triangles are congruent, the triangles are congruent
Angle-Angle-Side (AAS) Postulate If two angles and a non-included side are congruent to the corresponding two angles and a side of a second triangle, then the two triangles are congruent.
Angle-Angle Similarity Postulate Two corresponding angles of two or more triangles are congruent, the triangles are similar.
Angle-Angle-Side (AAS) Postulate If two angles and a non-included side are congruent to the corresponding two angles and a side of another triangle, the two triangles are congruent.
Angle-Side-Angle (AAS) Postulate If two angles and the included side of one triangle are congruent to two angles and an included side of another triangle, the triangles are congruent.
Arc Addition Postulate The measure of an arc created by two adjacent arcs can be found by adding the measures of two adjacent arcs
Area of a Square The area of the square is a measurement representing the spacer within the interior of a square. It is found by the formula A=Ssquared where a is the area and s is the length of aa side.
Area of a Triangle The area of a triangle is a measurement representing the space within the interior triangle. It is found by the formula A=1/2bh where a is the area, b is the base, and h is the length of the height.
Assoiative property of Addition For all real numbers, a,b, and c (a+b)+c equals (a+b)+C.
Assoiative Property of Multiplication For all real numbers, a, b, and c, a+(b+c)=(a+b)+c.
Associative Property of Multiplication For all real number a, b, and c, a times (b times c).
Commutative property of Addition For all real numbers a, b, and c, A+(b + c) equals (a + b) + c.
Associative Property of Multiplication For all real numbers a, b, and c, a time(b times c) equals (a time b) times c.
Communiative Property of Multiplication For all real numbers a and b, a times b equals b times a
Congruent Arcs and Chords Theorem Two minor arcs within a the same circle or between congruent circles are congruent if snd only if their corresponding chords are congruent.
Congruent Inscribed Angles Theorem Two or more inscribed angles that intercept the same arc, or congruent arcs, are congruent.
Corresponding Angles Postulate If a transversal intersects two parallel lines then the corresponding parts are congruent.
Cross Product Property For real numbers a, b, c and d, a divided by b equals c divided by d is equivalent to a tomes d equals b times c or ad equals bc.
Distance Between Two Points Postulate The distance between two points can be found by taking the absolute value of the difference between the coordinates of the two pints.
Distributive Property For all real numbers a, b, and c, a(b+c)= ab + ac.
Division Property of Equality For real numbers a, b, and c, if a=b and c is not equal to 0, then a divided by d equals b divided by c.
Exterior Angle to a Circle Theorem If two secants, two tangents, or a secant or a tangent intersect outside of a circle, the angle created between them is one half the absolute value of the difference of the measures of their intercepted arcs.
Hypotenuse-Leg (HL) Theorem If two right triangles have congruent hypotenuses and corresponding, congruent legs, the two right triangles are congruent.
Identity Property of Division Any number divided by one results in the same number, for example c divided by one equals c.
Inscribed Angle to a Semicircle Theorem An inscribed angle that intersects a semicircle is a right angle.
Inscribed Angle Theorem The measure of an inscribed angle is equal to one half the measure of its intercepted arc
Inscribed Quadrilateral Theorem The opposite angles of an inscribed quadrilateral to a circle are supplementary.
Intersecting Chords Theorem The point of intersection between two chords in a circle creates two pairs of segments whose products are equal.
Intersecting Lines Postulate If two lines intersect, then they intersect at exactly one point.
Isosceles Triangle Theorem If two sides are congruent, then the angles opposite those sides are congruent.
The Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent making the triangle an isosceles triangle.
Midsegmeent of a Triangle Theorem A segment connecting the two sides of a triangle is parallel to the third side and half its length.
Multiplication Property of Equality for real numbers a, b, and c, if a=b then ac = bc.
Multiplicative Identity Multiplying any number by 1 produces that number, for example b times 1 equals b
Opposite Angle Theorem If two angles of a triangle are not congruent, then the larger side lies opposite the larger angle.
Parallel Postulate Given a line and a point not on that line, there exists only one line through the given point parallel to the given line.
Perpendicular Diameter and Chords Theorem If a diameter is perpendicular to a chord, then the diameter bisects the chord and thee minor arc between the endpoints of the chord
Pieces of Right Triangles and Similarity Theorem If a altitude is drawn from the right angle of a right triangle is the geometric mean between the segments of the hypotenuse created by the intersection of the altitude and the hypotenuse.
Second Corollary to the Pieces of Right Triangles Similarity Theorem Each leg of a right triangle is the geometric mean between the hypotenuse and the segment of the hypotenuse created by the altitude adjacent to the given leg.
Pythagorean Theorem If a right triangle has sides a and b and hypotenuse c then a squared plus b squared equals c squared.
Converse of the Pythagorean Theorem In a triangle with sides a, b, and c if a squared plus b squared equals c squared, the triangles is a right triangle.
Reflexive Property of Equality for a real number a, a equals a
Same-side interior angles If a transversal intersects two parallel lines then the same-side interior angles are supplementary.
Converse of the same-side interior angles theorem If the same-side interior angles formed by two lines and a transversal are supplementary, then the two lines are parallel.
Secant-Tangent Intersection Theorem When a secant and a tangent intersects at the point of tangency, the angles created at the point of intersection are half the measurement of the arcs they intersect.
Segment Addition Postulate If point C is between points A and B, then AC plus CB equals AB.
Side-Angle-Side (SAS) Postulate If two sides and an included angle of one triangle are congruent to two sides and an included angle of another triangle, then the triangles are congruent.
Side-Angle-Side Similarity Postulate If two or more triangles have corresponding, congruent angles and the sides that make up these angles are proportional, then the triangles are similar.
Side-Side-Side (SSS) Postulate If the sides are congruent to the sides of a second triangle, then the triangles are congruent.
Side-Side-Side Similarity Theorem If three or more triangles have three corresponding, proportional sides, then the triangles are similar.
Square Root Property of Equality For any real number a, the square root of a squared equals a.
Substitution Property of Equality For real numbers a, b, and c if a equal b, then a minus c equals b minus c.
Supplementary Angles of a Trapezoid Theorem Consective angles between the two bases of a trapezoid are supplementary.
Symmetric Property of Equality For real numbers a and b, if a equals b, then b equals a.
Transitive Property of Equality For real numbers a, b, and c, if a equals b and b equals c, then a equals c.
Triangle Exterior Angle Theorem The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.
Triangle Inequality Theorem The sum of the two lengths of any two sides of a triangle is greater than the third side.
Corollary to the Triangle Inequality Theorem The length of the third side of the triangle is less than the total and greater the absolute value of the difference of the other two sides.
Triangle Proportionally Theorem If a line is parallel to one side of a triangle and also intersects the two sides, the line divides the sides proportionally.
Triangle Sum Theorem The sum of the measures of the angles in the triangle is 180 degrees.
Vertical Angles Theorem Vertical angles are congruent