constructions | definitions | theorems & postulates
(Note: The following are the most commonly used definitions in proofs. For a complete list of definitions for the course, refer to the glossary.)
Addition Property of Equality
For real numbers a, b, and c, if a = b, then a + c = b + c.
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 Exterior Angles Theorem
If a transversal intersects two parallel lines, then alternate exterior angles are congruent.
Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate interior angles are congruent.
Angle Addition Postulate
The measure of an angle created by two adjacent angles may 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 similar.
Angle-Angle-Side (AAS) Postulate
If two angles and a non-included side are congruent to the corresponding two angles and side of a second triangle, the two triangles are congruent.
Angle-Side-Angle (ASA) Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.
Arc Addition Postulate
The measure of an arc created by two adjacent arcs may be found by adding the measures of the two adjacent arcs.
Area of a Square
The area of a square is a measurement representing the space within the interior of a square. It is found by the formula A = s2 where A is the area and s is the length of a side.
Area of a Triangle
The area of a triangle is a measurement representing the space within the interior of a triangle. It is found by the formula A = bh where A is the area, b is the length of the base, and h is the length of the height.
Associative Property of Addition
For real numbers a, b, and c, a + (b + c) = (a + b) + c.
Associative Property of Multiplication
For real numbers a, b, and c, a • (b • c) = (a • b) • c.
Bisecting Diagonal Theorem
The diagonal of a kite connecting the vertex angles bisects the diagonal connecting the nonvertex angles.
Bisecting Vertex Angles Theorem
The diagonal of a kite connecting the vertex angles is an angle bisector of these vertex angles.
Central Angle Theorem
The measure of a central angle is equal to the measure of the arc it intercepts.
Commutative Property of Addition
For real numbers a and b, a + b = b + a.
Commutative Property of Multiplication
For real numbers a and b, a • b = b • a.
Concurrency of Altitudes Theorem
The lines containing the altitudes of a triangle are concurrent.
Concurrency of Angle Bisectors Theorem
The angle bisectors of a triangle are concurrent at a point equidistant from the sides of the triangle.
Concurrency of Medians Theorem
The medians of a triangle are concurrent at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side.
Concurrency of Perpendicular Bisectors Theorem
The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.
Congruent Arcs and Chords Theorem
Two minor arcs within the same circle or between congruent circles are congruent if and only if their corresponding chords are congruent.
Congruent Arcs Theorem
Two arcs are congruent if the central angles that intercept them are also congruent.
Congruent Inscribed Angles Theorem
Two or more distinct inscribed angles that intercept the same arc, or congruent arcs, are congruent.
Coplanar Points Postulate
Through any three non-collinear points, there is exactly one plane.
Corresponding Angles Postulate
If a transversal intersects two parallel lines, then corresponding angles are congruent.
Cross Product Property
For real numbers a, b, c, and d, is equivalent to a • d = b • c or ad = bc.
Diagonals of an Isosceles Trapezoid Theorem
The diagonals of an isosceles trapezoid are congruent.
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 points.
For 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 ≠ 0, then .
Exterior Angle to a Circle Theorem
If two secants, two tangents, or a secant and a tangent intersect outside a circle, the measure of the created angle between them is one-half the absolute value of the difference of the measures of their intercepted arcs.
Exterior Angle Sum Theorem
The sum of the exterior angles of any polygon total 360°.
The Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.
Hypotenuse-Leg (HL) Theorem
If two right triangles have congruent hypotenuses and corresponding, congruent legs, the two right triangles are congruent.
Inscribed Angle to a Semicircle Theorem
An inscribed angle that intercepts a semicircle is a right angle.
Inscribed Angle Theorem
The measure of an inscribed angle is equal to half the measure of its intercepted arc.
Inscribed Quadrilateral Theorem
The opposite angles of an inscribed quadrilateral to a circle are supplementary.
Interior Angle Sum Theorem
If n represents the number of sides in any polygon, the expression to find the sum of the interior angles in a polygon is (n – 2) • 180°.
Intersecting Lines Postulate
If two lines intersect, then they intersect in exactly one point.
Intersecting Planes Postulate
If two distinct planes intersect, then they intersect in exactly one line.
Isosceles Trapezoid Theorem
If the legs of a trapezoid are congruent, then the base angles are congruent.
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Midsegment of a Triangle Theorem
A segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
Midsegment of a Trapezoid Theorem
The length of the midsegment of a trapezoid is half of the total of the lengths of the bases.
Multiplication Property of Equality
For real numbers a, b, and c, if a = b, then ac = bc.
Nonvertex Angles Theorem
The nonvertex angles of a kite are congruent.
Opposite Angle Theorem
If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.
Opposite Side Theorem
If two sides of a triangle are not congruent, then the larger angle lies opposite the larger side.
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 Diagonals Theorem
The diagonals of a kite are perpendicular.
Perpendicular Diameters and Chords Theorem
If a diameter is perpendicular to a chord, then the diameter bisects the chord and the minor arc between the endpoints of the chord.
Pieces of Right Triangles Similarity Theorem
If an altitude is drawn from the right angle of a right triangle, the two smaller triangles created are similar to one another and to the larger triangle.
Through any two points there exists exactly one line.
Properties of Kites
Properties of Parallelograms
Properties of Rectangles
Properties of a Rhombus
Properties of a Square
Properties of a Trapezoid
Proportional Perimeter Theorem
If two triangles are similar, the perimeters of each triangle are proportional to their corresponding sides.
If a right triangle has sides a and b and hypotenuse c then a2 + b2 = c2.
Reflexive Property of Equality
For real number a, a = a.
Same-Side Interior Angles
If a transversal intersects two parallel lines, then same-side interior angles are supplementary.
Secant Interior Angle Theorem
The measure of a secant angle is equal to half the sum of the arcs it and its vertical angle intercept.
Secant-Tangent Intersection Theorem
When a secant and tangent intersect 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 + CB = AB.
Side-Angle-Side (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 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 of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.
Side-Side-Side Similarity Theorem
If two or more triangles have three corresponding, proportional sides, then the triangles are similar.
Square Root Property of Equality
For any real number a, = a.
Substitution Property of Equality
For real numbers a and b, if a = b, then a can replace b in any expression and vice versa.
Subtraction Property of Equality
For real numbers a, b, and c, if a = b, then a − c = b − c.
Supplementary Angles of a Trapezoid Theorem
Consecutive angles between the two bases of a trapezoid are supplementary.
Symmetric Property of Equality
For real numbers a and b, if a = b, then b = a.
Transitive Property of Equality
For real numbers a, b, and c, if a = b and b = c, then a = c.
Triangle Altitude Similarity Theorem
If two triangles are similar, the corresponding altitudes are proportional to each set of corresponding sides.
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 lengths of any two sides of a triangle is greater than the third side.
Triangle Proportionality Theorem
If a line is parallel to one side of a triangle and also intersects the other two sides, the line divides the sides proportionally.
Triangle Sum Theorem
The sum of the measures of the angles in a triangle is 180°.
Vertical Angles Theorem
Vertical angles are congruent.