Pairs of Angles

In geometry, pairs of angles can relate to each other in several ways. Some examples are complementary angles, supplementary angles, vertical angles, alternate interior angles,
alternate exterior angles, corresponding angles and adjacent angles.

Complementary Angles

Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees (right angle). One of the complementary angles is said to be the complement of the other.
The two angles do not need to be together or adjacent. They just need to add up to 90 degrees. If the two complementary angles are adjacent then they will form a right angle.

∠ABC is the complement of ∠CBD

Supplementary Angles

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

Vertical Angles

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 will require you to work out the values of angles given in diagrams by applying the relationships between the pairs of angles.

Example 1: Given the diagram below, determine the values of the angles x, y and z.

Solution:

Step 1: x is a supplement of 65°.

            Therefore, x + 65° =180° ⇒ x = 180° – 65° = 115°

Step 2: z and 115° are vertical angles.

            Therefore, z = 115°

Step 3: y and 65° are vertical angles.

            Therefore, y = 65°

Answer: x = 115°, y = 65° and z = 115°

Alternate Interior Angles

When a line intersects a pair of parallel lines alternate interior angles are 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 1: Given the diagram below, determine the values of the angles b, c, d, e, f, g and h.

Solution:

Step 1: b is a supplement of 60°.

            Therefore, b + 60° =180° ⇒ b = 180° – 60° = 120°

Step 2: b and c are vertical angles.

            Therefore, c = b = 120°

Step 3: d and 60° are vertical angles.

            Therefore, d = 60°

Step 4: d and e are alternate interior angles.

            Therefore, e = d = 60°

Step 5: f and e are supplementary angles.

            Therefore, f + 60° =180° ⇒ f = 180° – 60° = 120°

Step 6: g and f are vertical angles.

            Therefore, g = f = 120°

Step 7: h and e are vertical angles.

            Therefore, h = e = 60°

Answer: b = 120°, c = 120°, d = 60°, e = 60°, f = 120°, g = 120° and h = 60°

From the above example, you may notice that either an angle is 60° or it is 120°. Actually, all the small angles are 60° and all the big angles are 120°. In general, the diagram will be as shown below. The small and big pair of angles are supplementary (i.e. small + big = 180°). Therefore, given any one angle you would be able to work out the values of all the other angles.

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.

Corresponding Angles

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.

Adjacent angles

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.

Videos

The following video will give an overview of Complementary, Supplementary, and Vertical angle pairs.

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