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

A 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).

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

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 \(120°\).

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.

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 \(∠A\) as congruent to \(∠X\), we will write it as \(∠A ≅ ∠X\).

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.

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

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Frequently Asked Questions

How do you identify complementary, supplementary, vertical, adjacent and congruent angles?

Look at how the angles are placed and what their measures add up to. Adjacent angles share a vertex and a side and sit side by side. Vertical angles sit directly across from each other where two lines cross. Then check the numbers: a sum of 90° means complementary, a sum of 180° means supplementary, and equal measures mean congruent. One pair can fit more than one label, such as adjacent and supplementary.

What are the definitions of complementary, supplementary, vertical, adjacent and congruent angles?

Complementary angles add up to 90°, like 25° and 65°. Supplementary angles add up to 180°, like 130° and 50°. Vertical angles are the opposite angles formed when two lines intersect, and they are always equal. Adjacent angles share a common vertex and side without overlapping. Congruent angles have the same measure, written with the symbol ≅, as in ∠A ≅ ∠B.

How do you find complementary and supplementary angles?

Subtract from 90° to find a complement and from 180° to find a supplement. For an angle of 38°, the complement is 90° − 38° = 52° and the supplement is 180° − 38° = 142°. Notice that the supplement is always 90° more than the complement. An angle of 90° or more has no complement, because angle measures must be positive.

How do you find the measures of complementary, supplementary and vertical angles?

Write an equation from the relationship and solve it. If two vertical angles measure 3x + 5 and 5x − 25, set them equal: 3x + 5 = 5x − 25, so 30 = 2x and x = 15, which makes each angle 50°. If two complementary angles measure x and x + 20, write x + x + 20 = 90, so x = 35 and the angles are 35° and 55°.

Is there a complementary, supplementary, vertical, adjacent and congruent angles worksheet?

Yes. The exercises at the end of the lesson work as a short worksheet with answers, and Effortless Math also has printable angle worksheets. A good worksheet mixes the types, for example: find the supplement of 112° (68°), find the complement of 47° (43°), and find the angle vertical to a 76° angle (76°).

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• Types of angles
• How to measure angles
• Parallel and perpendicular lines
• How to find the central angle of a circle
• How to identify polygons