Pairs of Angles
In geometry, pairs of angles can relate to each other in several ways.
For example :
- complementary angles
- supplementary angles
- vertical angles
- alternate interior angles
- alternate exterior angles
- corresponding angles
- adjacent angles.
Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees (right angle).
The two angles do not need to be together or adjacent. When both the angles are added they must sum up to 90 degrees.
If the two complementary angles are adjacent then they will form a right angle.
∠ABC is the complement of ∠CBD
Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees (straight line).
One of the supplementary angles is said to be the supplement of the other.
The two angles do not need to be together or adjacent. They just need to add up to 180 degrees. If the two supplementary angles are adjacent then they will form a straight line.
∠ABC is the supplement of ∠CBD
Two pairs of angles are formed by two intersecting lines. Vertical angles are opposite
angles in such an intersection. Vertical angles are equal to each other.
Very often math questions require us to work out the values of angles given in diagrams by applying the relationships between the pairs of angles.
Alternate Interior Angles
When a line intersects a pair of parallel lines alternate interior angles is formed. Alternate interior angles are equal to each other.
One way to find the alternate interior angles is to draw a zigzag line on the diagram. In the above diagrams, d and e are alternate interior angles. Similarly, c and f are also alternate interior angles.
Example: Given the diagram below, determine the values of the angles b, c, d, e, f, g and h.
Alternate Exterior Angles
One way to remember alternate exterior angles is that they are the vertical angles of the alternate interior angles. Alternate exterior angles are equal to one another.
a and h are alternate exterior angles and they are equal to one another.
b and g are alternate exterior angles and they are equal to one another.
If two parallel lines are intersected by a transversal then alternate exterior angles are congruent.
When a line intersects a pair of parallel lines corresponding angles are formed. Corresponding angles are equal to each other.
One way to find the corresponding angles is to draw a letter F on the diagram. The F can also be facing the other way.
In the above diagram, d and h are corresponding angles.
There many other corresponding pairs of angles in the diagram:
b and f ; c and g ; a and e.
When two angles are next to one another, they are called adjacent angles. Adjacent angles share a common side and a common vertex.
Example: x and y are adjacent angles.