Proofs

Proofs

Definition of a Congruent

Segments or angles that have the same measure (Used when you switch from an "=" sign to a "≅" sign or vice versa)

Definition of a Bisector

a line or segment that intersects a segment or angle and divides it into two congruent halves

Definition of a Midpoint

divides the segment into two congruent segments

Angle Addition Postulate

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

Vertical Angles

A pair of opposite congruent angles formed by intersecting lines

Linear Pair Postulate

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

Reflexive Property

A line segment (or angle) is congruent to itself

Symmetric Property

a=b, b=a; 8=x, x=8

Transitive Property

a=b, b=c, a=c

Addition Property

The same thing can be added to both sides of an equation

Subtraction Property

The same thing can be subtracted from both sides of an equation

Multiplication Property

You can multiply both sides of an equation by the same thing

Division Property

You can multiply both sides of an equation by the same thing (except 0)

Substitution Property

Replace a variable or value with an equal variable or value

Distributive Property

a(b+c)=ab+ac

Definition of an Acute Angle

an angle that measures less than 90 degrees

Definition of an Obtuse Angle

An angle that measures more than 90 degrees but less than 180 degrees

Definition of a Right Angle

measures 90º

Right Angle Congruence Theorem

All right angles are congruent to each other

Definition of Perpendicular Lines

Two intersecting lines that form four right angles

Definition of Parallel Lines

Two lines are parallel if they do not intersect

Same-Side Interior Angles Theorem

Angles inside two parallel lines on the same side of the transversal are supplementary

Alternate Interior Angles Theorem

Angles inside two parallel lines on opposite sides of the transversal are congruent

Corresponding Angles

The two angles are in the same position at each parallel line. They are congruent.

Alternate Exterior Angles Theorem

Angles outside two parallel lines on opposite sides of the transversal are congruent

Triangle Sum Theorem

the sum of the measures of the angles of a triangle is 180

SSS Postulate

If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

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.

ASA Postulate

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

AAS Theorem

If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.

CPCTC (Both SSS & AAA)

If two triangles are congruent then all their corresponding parts are congruent. Used in a proof after showing triangles are congruent.

Definition of Isosceles Triangle

a triangle that has two congruent sides

Isosceles Triangle Theorem

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

Definition of a Parallelogram

If a quadrilateral is a parallelogram, then the quadrilateral's both pairs of opposite sides are parallel

(1/5 Characteristic of a Parallelogram)

Parallelogram sides

Opposite Sides are Parallel.

Opposite Sides are Congruent.

Parallelogram angles

Opposite Angles are Congruent.

Consecutive Angles are Supplementary.

Obtuse Triangle

A triangle with one angle that is greater than 90 degrees.