How to Find Complementary, Supplementary, Vertical, Adjacent, and Congruent Angles?

When angles appear in groups of two to display a certain geometrical property they are called angle pairs. There is a special relationship between pairs of angles. Some of the angle pairs contain complementary angles, supplementary angles, vertical angles, and adjacent angles.

Related Topics

• How to Find Complementary and Supplementary Angles
• How to Find Vertical Angles

Step by step guide to finding complementary, supplementary, vertical, adjacent, and congruent angles

When two angles are paired, then there exist different angles such as:

1. Complementary angles

If the sum of the two angles reaches \(90\) degrees, they are called complementary angles. In other words, when the complementary angles are placed together, they form a right angle (\(90\) degrees). In the figure given below, \(60° + 30° = 90°\). Hence, from the “definition of complementary angles”, these two angles are complementary to each other.

Each angle between the complementary angles is called the “complement” of the other angle. Here,

• \(60°\) is the complement of \(30°\).
• \(30°\) is the complement of \(60°\).

2. Supplementary angles

When the sum of the two angles is \(180\) degrees, they are called supplementary angles. The two angles when added make up \(180°\). For example, \(110°\) and \(70°\) make up \(180°\). So these two angles are said to be supplementary. Here, one angle is the supplement of another angle. For example, the supplement of \(60°\) is \((180° – 60°)\), which is \(120°\).

See the figure below where \(130° + 50° = 180°\). Hence, by defining supplementary angles, these two angles are supplementary.

The difference between complementary angles and supplementary angles are given in the table below:

3. Vertical angles

Vertical angles are a pair of non-adjacent angles formed by the intersection of two straight lines. In simple words, vertical angles are located across from one another in the corners of the \(“X”\) formed by two straight lines. They are also called vertically opposite angles as they are situated opposite each other.

4. Adjacent angles

Two angles are called adjacent angles if they have a common vertex, a common side, and no overlap. See the figure below to see what the adjacent angles are like. Angle \(1\) and \(2\) are adjacent because they have a common side \(BD\) and a common vertex \(B\).

Congruent angles

The definition of congruent angles is “angles that are equal in the measure are known as congruent angles”. In other words, equal angles are congruent angles. It is denoted by the symbol \(“≅”\), so if we want to represent \(∠A\) is congruent to \(∠X\), we will write it as \(∠A ≅ ∠X\). Look at a congruent angles example given below.

In the image above, both angles are equal in measurement (\(60^∘\) each). They can completely overlap each other. So, as per the definition, we can say that both the given angles are congruent.

Finding Complementary, Supplementary, Vertical, Adjacent, and Congruent Angles – Example 1:

Find the angle \(x\) in the following figure.

Solution:

In the given figure, \(x\) and \(62°\) are complementary angles as they form a right angle. Hence, their sum is 90°.

\(x + 62° = 90°\)

\(x = 90°-62°\)

Therefore, the value of angle \(x\) is \(28°\).

Exercises for Finding Complementary, Supplementary, Vertical, Adjacent, and Congruent Angles

• Find the measurement of angle \(f\).

• Find the measures of angles \(x\), \(y\), and \(a\) in the figure.

• \(\color{blue}{23°}\)
• \(\color{blue}{∠x = 75°, ∠y = 105°, ∠a =75°}\)