Glossary of Theorems and Postulates


Commonly used

Given

A piece of information is either explicitly mentioned in the given statement or is obvious in the diagram

Definition

Part of a geometric defintion (i.e. a rectangle has four right angles because that's how it is defined).

Algebra

Follows one of the properties of algebra. This is mostly used as a reason for simplifying or solving an equation.

Substitution

Take one quantity that is equal to another and "plug it in".

CPCTC

"Corresponding parts of congruent triangles are congruent". This is the deifintion of congruent triangles. If they are congruent, all their matching sides and angles are equal.


Lines and Angles

Two points determine a line

If you have two points, you can draw a line through them. Usually used as a reason when drawing a line between two points in a diagram.

Ruler postulate

Subtract the coordinates of two points to measure the distance between them.

Protractor postulate

Subtract the coordinates of the two rays of an angle to measure the angle.

Segment addition theorem

A line segment is equal to the sum of its parts (the smaller segments that make it up).

Angle addition theorem

An angle is equal to the sum of its parts (the smaller angles that make it up).


Congruence

Angle-Side-Angle (ASA)

If two angles and the side between them are equal in both triangles, the triangles are congruent.

Side-Angle-Side (SAS)

It two sides and the angle between them are equal in both triangles, the triangles are congruent.

Side-Side-Side (SSS)

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

Isosceles triangle theorems

If two sides of one triangle are equal so are their opposite angles.

If two angles of one triangle are equal so are their opposite sides.


Inequalities

Exterior angle theorem

The exterior angle of a triangle is greater than either of its remote interior angles (the angles inside the triangle that the exterior angle is not in a linear pair with).

Unequal angles and sides theorems

If two angles of a triangle have different measures, the side across from the bigger angle is larger than the side across from the smaller angle. In other words, bigger angle are across from bigger sides and smaller angles are across from smaller sides.

Triangle inequality theorem

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


Quadrilaterals

Quadrilateral angle sum theorem

The sum of the interior angles of a quadrilateral is 360 degrees.

The opposites sides and angles of a parallelogram are equal

If a quadrilateral is a parallelogram, both pairs of opposite sides and angles are equal to each other.

The diagonals of a parallelogram bisect each other

If a quadrilateral is a parallelogram, the point where the diagonals meet divides each diagonal in half.

Determining if a quadrilateral is a parallelogram

A quadrilateral is a parallelogram if any of the following conditions is met: both pairs of opposite sides are equal, both pairs of opposite angles are equal, the diagonals bisect each other, or one pair of opposite sides is parallel and equal.

Midsegment theorem

If you connect the midpoints of two sides of a triangle, it forms a midsegment. That midsegment is half as long as the third side of the triangle and also parallel to it.


Area

Area postulate

The area postulate says two things: (1) congruent triangles must have equal areas and (2) you can add up all the non-overlapping regions of a polygon to calculate its area.

Converse of the Pythagorean theorem

If the lengths of the three sides of a triangle are a solution to the equation of the Pythagorean theorem, then the triangle must be a right triangle.


Similarity

Side-Splitter theorem

If a line intersects two sides of a triangle and is parallel to the third side, that line divides the sides it intersects into lengths that form the same ratio.

AA similarity

Two triangles are similar to each other if they share two equal angles.