Vocabulary and basic measures
Definition and special angles
Angle
An angle is a shape formed by two line segments or rays stemming from a common point called the vertex.
We can label the above angle as either \angle ABC or \angle CBA.
Angles are measured in degrees so that a full rotation is 360 (360^\circ). In particular:
- A quarter rotation is 90^\circ.
- A half rotation is 180^\circ.
Acute angle
An acute angle is any angle less than 90^\circ.
In the above graphic, we have \angle ABC \lt 90^\circ.
Obtuse angle
An obtuse angle is any angle over 90^\circ.
In the above graphic, we have \angle ABC \gt 90^\circ.
Straight angle
A straight angle is an angle of 180^\circ.
In the above graphic, we have \angle ABC = 180^\circ.
Right angle
A right angle is an angle of 90^\circ.
In the above graphic, we have \angle ABC = 90^\circ.
Reflex angle
A reflex angle is an angle larger than 180^\circ.
Complementary, supplementary and congruent angles
Complementary angles
Two angles are said to be complementary if their sum is 90^\circ.
The angles 30^\circ and 60^\circ are complementary since:
30^\circ +60^\circ=90^\circ
In the following graphic, the angles \alpha and \beta are complementary.
Supplementary angles
Two angles are said to be supplementary if their sum is 180^\circ.
The angles 120^\circ and 60^\circ are supplementary because:
120^\circ + 60^\circ =180^\circ
In the following graphic, the angles \alpha and \beta are supplementary.
Congruent angles
Two angles are said to be congruent if they have the same measurement in degrees.
Any two angles measuring 30^\circ are congruent.
In the following graphic, the angles \alpha and \beta are congruent.
Special angles and their measures
Adjacent and vertical angles
Adjacent angles
Two angles are said to be adjacent if they share a common side and have the same vertex.
In the following graphic, the angles \alpha and \beta are adjacent.
The sum of two adjacent angles is equal to the measurement of the larger angle formed by the two angles.
Vertical angles
Vertical angles are a pair of opposite angles formed by two intersecting lines.
In the following graphic, the angles \alpha and \beta form a pair of vertical angles. The angles \delta and \epsilon also form a pair of vertical angles.
A pair of vertical angles are always congruent. In the following graphic, the angles \alpha and \beta are congruent and \delta and \epsilon are congruent.
Corresponding angles
Corresponding angles
Suppose two lines are intersected by a third line. The angles in matching corners are called corresponding angles.
If two parallel lines are intersected by a third line, then any pair of corresponding angles are congruent.
In the following graphic, the angles \alpha and \beta are congruent.
Alternate interior and alternate exterior angles
Alternate interior angles
Suppose two lines are intersected by a third line. Then any pair of angles formed on the opposite side of the third line, but on the interior of the original two lines, are called alternate interior angles.
In the following graphic, the angles \alpha and \beta are alternate interior angles.
If a pair of parallel lines are intersected by a third line, then any pair of alternate interior angles are congruent.
In the following graphic, the angles \alpha and \beta are congruent.
Alternate exterior angles
Suppose two lines are intersected by a third line. Then any pair of angles on opposite sides of the third line and on the exterior of the original two lines are called alternate exterior angles.
In the following graphic, the angles \alpha and \beta are alternate exterior angles.
If a pair of parallel lines are intersected by a third line, then any pair of alternate exterior angles are congruent.
In the following graphic, the angles \alpha and \beta are congruent.
The following graphic contains two congruent exterior angles of measure 45^\circ.
Same-side interior angles
Same-side interior angles
Suppose two lines are intersected by a third line. Then a pair of same-side interior angles are any two angles on the same side as the third line, but on the interior of the original two lines.
In the following graphic, the angles \alpha and \beta are same-side interior angles.
Suppose two parallel lines are intersected by a third line. Then any pair of same-side interior angles are supplementary.
The following graphic contains two same side interior angles which are supplementary. One of the angles is 45^\circ and the other is 135^\circ.
Bisector
Bisector
A bisector is either a line, line segment, or ray which splits an angle into two congruent angles.
Proofs involving angles
The above properties of angles allow us to find new properties involving angles.
In the following graphic, two parallel lines are intersected by a third line.
We can prove that the angles \alpha and \beta are supplementary. To do this, we first consider two other angles, \delta and \epsilon, as labeled in the following graphic.
The angles \alpha and \delta form a pair of vertical angles and are therefore congruent. Similarly, the angles \beta and \epsilon form a pair of vertical angles and are therefore congruent. Thus, we have the following two pieces of information:
- \alpha=\delta
- \beta=\epsilon
The angles \delta and \epsilon form a pair of same-side interior angles and are therefore supplementary. Hence we also know that:
- \delta+\epsilon=180^\circ
By substituting \alpha in for \delta and \beta in for \epsilon, we find that:
\alpha+\beta=180^\circ
Therefore, the angles \alpha and \beta are indeed supplementary.