Postulates and Theorems - Geometry

These flashcards cover essential postulates, theorems, definitions, and properties relevant to geometry, useful for exam preparation.

Postulate 1-1

Through any two points, there is exactly one line.

Postulate 1-2

If two distinct lines intersect, then they intersect in exactly one point.

Postulate 1-3

If two distinct planes intersect, they intersect in exactly one line.

Postulate 1-4

Through any three noncollinear points, there is exactly one plane.

Ruler Postulate

Every point on a line can be paired with a real number.

Protractor Postulate

Every ray can be paired one to one with a real number from 0 to 180.

Linear Pair Postulate

If two angles form a linear pair, then they are supplementary.

Law of Detachment

If p→q is true and p is true, then q is true.

Symmetric Property

If AB = CD, then CD = AB.

Reflexive Property

For any segment AB, AB = AB.

Segment Addition Postulate

If B is between A and C, then AB + BC = AC.

Angle Addition Postulate

If point B is in the interior of angle AOC, then m∠AOB + m∠BOC = m∠AOC.

Law of Syllogism

If p→q is true and q→r is true, then p→r is true.

Vertical Angles Theorem

Vertical angles are congruent.

Congruent Complements Theorem

If two angles are complements of the same angle, then they are congruent.

Transitive Property

If AB = CD and CD = EF, then AB = EF.

Congruent Supplements Theorem

If two angles are supplements of the same angle, then they are congruent.

Parallel Postulate

Through a point not on a line, there is one and only one line parallel to the given line.

Perpendicular Postulate

Through a point not on a line, there is one and only one line perpendicular to the given line.

Triangle Angle-Sum Theorem

The sum of the measures of the angles of a triangle is 180 degrees.

Triangle Exterior Angle Theorem

The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles.

Slopes of Parallel Lines

If two non-vertical lines are parallel, then their slopes are equal.

Slopes of Perpendicular Lines

If two non-vertical lines are perpendicular, then the product of their slopes is -1.

Definition of Midpoint

A midpoint divides the segment into two congruent segments.

Definition of Segment Bisector

A segment bisector intersects a segment at its midpoint.

Definition of Complementary Angles

Two angles are complementary if the sum of their measures is 90 degrees.

Definition of Right Angle

A right angle measures 90 degrees.

Definition of Angle Bisector

An angle bisector divides an angle into two congruent angles.

Definition of Supplementary Angles

Two angles are supplementary if the sum of their measures is 180 degrees.

Definition of Congruent Angles

Congruent angles have the same measure.

Definition of Congruent Segments

Congruent segments have the same length.

Definition of Isosceles Triangle

An isosceles triangle has at least two congruent sides.

Overlapping Segment Theorem

If two collinear segments are adjacent to a common segment and congruent, then the overlapping segments are congruent.

Overlapping Angle Theorem

If two angles are adjacent to a common angle and congruent, then the overlapping angles are congruent.

Third Angles Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

Side-Side-Side (SSS) Postulate

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

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, the triangles are congruent.

Angle-Angle-Side (AAS) Theorem

If two angles and a non-included side of one triangle are congruent to those of another triangle, the triangles are congruent.

Angle-Side-Angle (ASA) Postulate

If two angles and the included side of one triangle are congruent to those of another triangle, the triangles are congruent.

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Corollary of Equilateral Triangle

If a triangle is equilateral, then it is also equiangular.

Hypotenuse-Leg (HL) Theorem

If the hypotenuse and a leg of one right triangle are congruent to those of another right triangle, then the triangles are congruent.

Polygon Angle-Sum Theorem

The sum of the measures of an n-gon is (n-2) × 180.

Polygon Exterior Angle-Sum Theorem

The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

Parallelogram Opposite Sides Theorem

If a quadrilateral is a parallelogram, then its opposite sides are congruent.

Parallelogram Consecutive Angles Theorem

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

Parallelogram Opposite Angles Theorem

If a quadrilateral is a parallelogram, then its opposite angles are congruent.

Parallelogram Diagonals Theorem

If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Rhombus Diagonal Perpendicularity Theorem

If a parallelogram is a rhombus, then its diagonals are perpendicular.

Rhombus Angles Theorem

If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

Rectangle Diagonal Theorem

If a parallelogram is a rectangle, then its diagonals are congruent.

Square Diagonal Theorem

If a quadrilateral is a parallelogram with perpendicular, congruent diagonals, then it is a square.

Isosceles Trapezoid Base Angles Theorem

If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent.

Trapezoid Midsegment Theorem

The midsegment of a trapezoid is parallel to the bases and half the sum of their lengths.

Triangle Midsegment Theorem

A segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long.

Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.

Concurrency of Angle Bisectors Theorem

The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

Concurrency of Medians Theorem

The medians of a triangle are concurrent at a point two-thirds of the distance from each vertex to the midpoint of the opposite side.

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is opposite the larger included angle.

Converse of the Hinge Theorem

If the third sides of two triangles are not congruent and the included angles are not congruent, then the larger included angle is opposite the larger third side.