GEO THEROMS

Ruler Postulate

Points on a line can be matched one-to-one with real numbers.

Segment Addition Postulate

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

Protractor Postulate

Rays can be matched one-to-one with numbers from 0° to 180°.

Angle Addition Postulate

If P is inside an angle, the whole angle equals the sum of its parts.

Two Point Postulate

Through any two points there is exactly one line.

Line-Point Postulate

A line contains at least two points.

Line Intersection Postulate

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

Three Point Postulate

Through any three noncollinear points there is exactly one plane.

Plane-Point Postulate

A plane contains at least three noncollinear points.

Plane-Line Postulate

If two points lie in a plane, the line containing them lies in the plane.

Plane Intersection Postulate

If two planes intersect, their intersection is a line.

Linear Pair Postulate

Angles that form a linear pair are supplementary.

Parallel Postulate

Through a point not on a line, exactly one parallel line can be drawn.

Perpendicular Postulate

Through a point not on a line, exactly one perpendicular line can be drawn.

Translation Postulate

A translation is a rigid motion.

Reflection Postulate

A reflection is a rigid motion.

Rotation Postulate

A rotation is a rigid motion.

Arc Addition Postulate

The measure of a larger arc equals the sum of adjacent arcs.

Reflexive Property of Segment Congruence

A segment is congruent to itself.

Symmetric Property of Segment Congruence

If AB ≅ CD, then CD ≅ AB.

Transitive Property of Segment Congruence

If AB ≅ CD and CD ≅ EF, then AB ≅ EF.

Reflexive Property of Angle Congruence

An angle is congruent to itself.

Symmetric Property of Angle Congruence

If ∠A ≅ ∠B, then ∠B ≅ ∠A.

Transitive Property of Angle Congruence

If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.

Right Angles Congruence Theorem

All right angles are congruent.

Congruent Supplements Theorem

Supplements of the same angle are congruent.

Congruent Complements Theorem

Complements of the same angle are congruent.

Vertical Angles Congruence Theorem

Vertical angles are congruent.

Corresponding Angles Theorem

Parallel lines cut by a transversal create congruent corresponding angles.

Alternate Interior Angles Theorem

Parallel lines create congruent alternate interior angles.

Alternate Exterior Angles Theorem

Parallel lines create congruent alternate exterior angles.

Consecutive Interior Angles Theorem

Consecutive interior angles are supplementary.

Corresponding Angles Converse

Congruent corresponding angles imply parallel lines.

Alternate Interior Angles Converse

Congruent alternate interior angles imply parallel lines.

Alternate Exterior Angles Converse

Congruent alternate exterior angles imply parallel lines.

Consecutive Interior Angles Converse

Supplementary consecutive interior angles imply parallel lines.

Transitive Property of Parallel Lines

Lines parallel to the same line are parallel.

Linear Pair Perpendicular Theorem

If two lines form congruent adjacent angles, they are perpendicular.

Perpendicular Transversal Theorem

A transversal perpendicular to one of two parallel lines is perpendicular to the other.

Lines Perpendicular to a Transversal Theorem

Two lines perpendicular to the same line are parallel.

Slopes of Parallel Lines

Parallel lines have equal slopes.

Slopes of Perpendicular Lines

Perpendicular slopes are negative reciprocals.

Triangle Sum Theorem

Interior angles of a triangle add to 180°.

Corollary to the Triangle Sum Theorem

Acute angles in a right triangle are complementary.

Exterior Angle Theorem

An exterior angle equals the sum of the two remote interior angles.

Third Angles Theorem

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

SAS Congruence Theorem

Two sides and the included angle congruent means triangles are congruent.

SSS Congruence Theorem

Three sides congruent means triangles are congruent.

HL Congruence Theorem

In right triangles, congruent hypotenuse and leg means triangles are congruent.

ASA Congruence Theorem

Two angles and included side congruent means triangles are congruent.

AAS Congruence Theorem

Two angles and a nonincluded side congruent means triangles are congruent.

Perpendicular Bisector Theorem

Points on a perpendicular bisector are equidistant from segment endpoints.

Converse Perpendicular Bisector Theorem

Equidistant points lie on the perpendicular bisector.

Angle Bisector Theorem

Points on an angle bisector are equidistant from the sides.

Converse Angle Bisector Theorem

Equidistant points from angle sides lie on the angle bisector.

Circumcenter Theorem

The circumcenter is equidistant from all vertices.

Incenter Theorem

The incenter is equidistant from all sides.

Centroid Theorem

The centroid divides each median in a 2:1 ratio.

Triangle Midsegment Theorem

A midsegment is parallel to the third side and half its length.

Triangle Longer Side Theorem

The longer side is opposite the larger angle.

Converse Triangle Longer Side Theorem

The larger angle is opposite the longer side.

Triangle Inequality Theorem

The sum of any two sides is greater than the third side.

Hinge Theorem

Larger included angle means longer third side.

Converse Hinge Theorem

Longer third side means larger included angle.

Polygon Interior Angles Theorem

Sum of interior angles = (n - 2) × 180.

Corollary to the Polygon Interior Angles Theorem

Sum of exterior angles of a convex polygon = 360°.

Polygon Exterior Angles Theorem

The exterior angles of a convex polygon add to 360°.

Parallelogram Opposite Sides Theorem

Opposite sides of a parallelogram are congruent.

Parallelogram Opposite Angles Theorem

Opposite angles of a parallelogram are congruent.

Parallelogram Consecutive Angles Theorem

Consecutive angles of a parallelogram are supplementary.

Parallelogram Diagonals Theorem

Diagonals of a parallelogram bisect each other.

Parallelogram Opposite Sides Converse

If both pairs of opposite sides are congruent, the quadrilateral is a parallelogram.

Parallelogram Opposite Angles Converse

If both pairs of opposite angles are congruent, the quadrilateral is a parallelogram.

Opposite Sides Parallel and Congruent Theorem

If one pair of opposite sides is both parallel and congruent, the figure is a parallelogram.

Parallelogram Diagonals Converse

If diagonals bisect each other, the quadrilateral is a parallelogram.

Rhombus Corollary

A quadrilateral is a rhombus if and only if all four sides are congruent.

Rectangle Corollary

A quadrilateral is a rectangle if and only if it has four right angles.

Square Corollary

A quadrilateral is a square if and only if it is both a rhombus and a rectangle.

Rhombus Diagonals Theorem

Diagonals of a rhombus are perpendicular.

Rhombus Opposite Angles Theorem

Each diagonal of a rhombus bisects opposite angles.

Rectangle Diagonals Theorem

Diagonals of a rectangle are congruent.

Isosceles Trapezoid Base Angles Theorem

Each pair of base angles is congruent.

Isosceles Trapezoid Base Angles Converse

Congruent base angles imply an isosceles trapezoid.

Isosceles Trapezoid Diagonals Theorem

Diagonals of an isosceles trapezoid are congruent.

Trapezoid Midsegment Theorem

Midsegment is parallel to the bases and has length (base1 + base2)/2.

Kite Diagonals Theorem

Diagonals of a kite are perpendicular.

Kite Opposite Angles Theorem

A kite has one pair of congruent opposite angles.

Perimeters of Similar Polygons Theorem

Ratio of perimeters equals ratio of corresponding side lengths.

Areas of Similar Polygons Theorem

Ratio of areas equals the square of the ratio of corresponding sides.

AA Similarity Theorem

Two congruent angles imply similar triangles.

SSS Similarity Theorem

Corresponding sides proportional imply similar triangles.

SAS Similarity Theorem

Two proportional sides and an included congruent angle imply similar triangles.

Triangle Proportionality Theorem

A line parallel to one side divides the other two sides proportionally.

Converse Triangle Proportionality Theorem

Proportional side segments imply the line is parallel to the third side.

Three Parallel Lines Theorem

Three parallel lines cut transversals proportionally.

Triangle Angle Bisector Theorem

An angle bisector divides the opposite side proportionally.

Pythagorean Theorem

a² + b² = c².

Converse of the Pythagorean Theorem

If a² + b² = c², then the triangle is a right triangle.