Math Terms and Definitions - Proofs

Adjacent Angles

Angles that share a vertex and a common side. They cannot share interior angles.

Midpoint

A point that divides a segment into two congruent segments.

Congruent

Same size, same shape.

Congruent Segments

Segments with the same lengths.

Congruent Angles

Angles with the same measure.

Collinear

Points that are present on the same line.

Non-Collinear

Points that are not present on the same line.

Coplanar

Points that are on the same plane.

Non-Coplanar

Points that do not lie on the same plane.

Between

A point between two distinct points.

Segment

A part of a line with two endpoints, that includes all points in between.

Linear Pair

Two adjacent angles that do not share a common vertex or side. These form a line.

Angle Bisector

A ray in the interior of the angle that forms two congruent angles.

Postulate

A statement accepted as true without proof.

Theorem

A proven statement.

Line Postulate

Through any two points, there is exactly one line.

Parallel Postulate

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

Perpendicular Postulate

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

Linear-Pair Postulate

If two linear pairs create a line, they are supplementary. (You need to establish that the angles form a linear pair.)

Segment Addition Postulate

If point B is collinear with and between points A and C, then AB + BC = AC. (You must identify points A, B, and C, confirming their collinearity.)

Angle Addition Postulate

If point D is in the interior of ∠ABC, then m∠ABD + m∠DBC = m∠ABC. (You need to confirm that point D lies in the interior.)

Reflexive Property

Equality a = a.

Congruency

Line AB is congruent to line AB.

Symmetric Property

Equality if a = b, then b = a.

Transitive Property

Equality if a = b, and b = c, then a = c.

Substitution Property

If a = b, then b can be replaced by a and vice versa.

Distributive Property

a(b + c) = ab + ac.

Addition Property

Equality if a = b, then a + c = b + c.

Subtraction Property

Equality if a = b, then a - c = b - c.

Multiplication Property

Equality if a = b, then ac = bc.

Division Property

Equality if a = b, then a/c = b/c.

Median

A segment drawn from a vertex to the opposite side.

Midsegments

Segments that connect the midpoints of two sides of a triangle.

Midsegment Theorem

The length of a midsegment is always half of the base of the triangle. It is also parallel to the base of the triangle. (You must identify the triangle and the midsegment.)

Parallelogram

A quadrilateral with two pairs of parallel sides.

Rectangle

An equiangular parallelogram. All the angles measure 90 degrees.

Diagonals

Segments that connect opposite vertices in a parallelogram.

Corresponding Angles

Angles that are in the same position on two parallel lines cut by a transversal.

Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. (You need to establish that the lines are parallel.)

Converse of Corresponding Angles Postulate

If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel. (You need to declare that they are cut by a transversal and that angles are congruent.)

AA Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. (You need to confirm the angle measurements.)

Linear Pair Postulate

If two angles form a linear pair, they are supplementary. (You must declare the angles are adjacent.)

Alternate Interior Angle Theorem

If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. (You need to establish that the lines are parallel.)

Alternate Exterior Angle Theorem

If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent. (You need to establish that the lines are parallel.)

Same-Side Interior Angle Theorem

If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary. (You need to establish that the lines are parallel.)

Same-Side Exterior Angle Theorem

If two parallel lines are cut by a transversal, then the pairs of same-side exterior angles are supplementary. (You need to establish that the lines are parallel.)

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. (You must identify the exterior angle and the relevant interior angles.)

Quadrilateral Sum Theorem

The sum of the interior angles of a quadrilateral is always 360 degrees. (You need to confirm the quadrilateral's shape.)

Triangle Sum Theorem

The sum of the interior angles of a triangle is always 180 degrees. (You need to confirm the triangle's shape.)

Right Triangle Complements Theorem

The two non-right angles in a right triangle are complementary (their measures add up to 90 degrees). (You must establish that one angle is a right angle.)

Converse to the Same Side Interior Angle Theorem

If two lines are cut by a transversal and the same-side interior angles are supplementary, then the lines are parallel. (You need to confirm the supplementary angles and declare the lines being cut by the transversal.)

Converse to the Same Side Exterior Angle Theorem

If two lines are cut by a transversal and the same-side exterior angles are supplementary, then the lines are parallel. (You need to confirm the supplementary angles and declare the lines being cut by the transversal.)

Congruent Supplements Theorem

If two angles are supplementary to the same angle (or congruent angles), then they are congruent to each other. (You must declare the angles that are supplementary to the same angle.)

Congruent Complements Theorem

If two angles are complementary to the same angle (or congruent angles), then they are congruent to each other. (You must declare the angles that are complementary to the same angle.)

Linear Pair Theorem

If two angles form a linear pair, they equal 180 degrees. (You need to declare that they form a linear pair.)

Right Angle Theorem

All right angles are congruent (they all measure 90 degrees).

Vertical Angles Theorem

Vertical angles, formed when two lines intersect, are congruent. (You need to declare that the angles are vertical.)

if perpendicular —> congruent adjacent angles

if two lines are perpendicular, then their adjacent angles are congruent

if congruent adjacent angles, then perpendicular

if adjacent angles are congruent, then the two lines are perpendicular

congruent segments theorem (biconditional)

AB is congruent to CD if and only if AC is congruent to BD

congruent angle theorem (biconditional)

<AOC is congruent to <BOD if and only if <AOB is congruent to <COD

parallel-parallel theorem

if two lines are parallel to the same line, then they are parallel

double perpendicular theorem

in the same plane, if two lines are perpendicular to the same line, then they are parallel