Key Theorems and Properties in Geometry

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.

Addition Property of Equality

The same value or expression can be added to both sides of an equation without changing its equality.

Alternate Exterior Angles Theorem

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

Alternate Interior Angles Theorem

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

All Right Angles are congruent to each other

A theorem stating that any two right angles will always have the same measure (90 degrees).

Angle Bisector

A ray that divides an angle into two equal, congruent angles.

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.

Bisector

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

Congruent Angles

Angles that have the same measure.

Congruent Segments

Segments or angles that have the same measure (Used when you switch from an '=' sign to a '≅' sign or vice versa). In other context, segments with equal lengths.

Corresponding Angles Theorem

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

CPCTC

Abbreviation for 'Corresponding Parts of Congruent Triangles are Congruent.' Used in a proof after showing triangles are congruent to prove specific corresponding parts are congruent.

Definition of a Parallelogram

If a quadrilateral is a parallelogram, then both pairs of its opposite sides are parallel.

Definition of a Right Angle

An angle that measures exactly 90º.

Definition of an Isosceles Triangle

A triangle that has two congruent sides.

Definition of Complementary Angles

Two angles whose degree measures have a sum of 90º.

Definition of Perpendicular Lines

Two intersecting lines that form four right angles.

Definition of Supplementary Angles

Two angles whose degree measures have a sum of 180º.

Distributive Property of Equality

A property where a(b+c) = ab+ac.

Equilateral Triangle

A triangle that is equilateral (all sides equal) if and only if it is equiangular (all angles equal). Each angle measures 60 degrees.

Hypotenuse Leg Theorem (HL)

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

If Two Angles are Congruent and Supplementary

Then each of those angles is a right angle.

Isosceles Triangle Theorem

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

Linear Pair

Two adjacent angles that form a straight line and are supplementary.

Median (of a triangle)

A segment in a triangle that connects the vertex of an angle to the midpoint of the opposite side.

Midpoint

A point that divides a segment into two congruent segments.

Multiplication Property of Equality

Both sides of an equation can be multiplied by the same non-zero number without changing its equality.

Parallelogram Characteristics

Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other.

Perpendicular Lines

Lines that intersect to form right angles.

Quadrilateral with bisecting and congruent diagonals

A rectangle.

Quadrilateral with bisecting and perpendicular diagonals

A rhombus.

Quadrilateral with bisecting, perpendicular, and congruent diagonals

A square.

Reflexive Property (of Equality)

A line segment (or angle) is congruent to itself.

Same Side Interior Angles Postulate

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

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.

SAS~ (Similarity)

Two triangles are similar if two corresponding sides are proportional and the included angle is congruent.

Segment Bisector

A line that divides a segment into two equal parts.

SSS Postulate

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

SSS~ (Similarity)

Two triangles are similar if their corresponding sides are proportional.

Substitution Property of Equality

A variable or value can be replaced with an equal variable or value in an equation.

Subtraction Property of Equality

The same value or expression can be subtracted from both sides of an equation without changing its equality.

Third Angles Theorem

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

Transitive Property of Equality

If a=b and b=c, then a=c.

Transversal

A line that intersects two or more lines.

Triangle Sum Theorem

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

Two lines parallel to the same line

These two lines are parallel to each other.

Vertical Angles

Angles opposite each other when two lines intersect; they are always congruent and can be assumed from a picture.