Definitions and postulates

Measuring angles

Classically, we consider a full rotation to be \(360^\circ\). This is when the terminal arm of an angle rotates until it comes back to the initial arm. Therefore, one degree is considered \(\frac{1}{360}\) of a full rotation.

Classification of angles

Related angles

Two angles that share an arm (but are on two different sides of that arm) are called adjacent angles. Two angles that add up to \(90^\circ\) are called complementary angles. Two angles that add up to \(180^\circ\) are called supplementary angles.

Intersecting lines

When two lines intersect, they create two pairs of opposite (also called vertically opposite), equal angles. Two lines that make an angle of \(90^\circ\) create four right angles; such lines are said to be perpendicular lines.

Parallel lines

Parallel lines in the same plane never intersect. If they are intersected by a third line, the resulting interior angles (see below) add up to \(180^\circ\). The shortest (perpendicular) distance between any two parallel lines is constant everywhere.

Transversal of parallel lines

A transversal is a a line that intersects two other (not necessary parallel) lines, creating 8 angles. The transversal of two parallel lines is interesting because the created angles are either congruent (equal) or supplementary. They are given particular names.

Congruent and similar triangles

Similar triangles look like each other, but they are not necessarily copies of one another. Two triangles are similar if and only if

their corresponding angles are congruent; or
the ratios of their corresponding sides are equal.

Congruent triangles are exact copies of one another. Two triangles are congruent if and only if

their corresponding sides are congruent;
two pairs of corresponding angles are congruent and a pair of corresponding sides are congruent; or
two pairs of corresponding sides are congruent and the angles enclosed by those sides are also congruent.