Complementary Angles- Definition, Types, Examples

What are Complementary Angles?

To understand what are complementary angles, we first need to remember what a Right Angle is. A right angle looks like the letter "L" and measures exactly 90°.

The complementary angles definition is very simple:

Complementary angles are a pair of angles whose sum is exactly 90°.

If you have two angles, and when you add their measurements together you get 90°, they are "complements" of each other. For example, if Angle A is 30° and Angle B is 60°, they are complementary because $30 + 60 = 90$.

Meaning of Complementary Angles

The Latin term complementum, which meaning "that which fills up," is where the word "complementary" derives from. In geometry, one angle "fills up" the space that the other angle needs to make a right angle.

Important Rules to Remember:

• Two is the Magic Number: Complementary angles always come in pairs. You cannot have three complementary angles that add up to 90°.

• Positive Only: In basic geometry, complementary angles must be positive.

• Acute Only: Since their sum is 90°, each individual angle must be less than 90°. This means both angles in a complementary pair are always acute angles.

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Types of Complementary Angles 

Not all pairs of complementary angles look the same. There are two main complementary angles types based on how they are positioned.

A. Adjacent Complementary Angles

"Adjacent" is a fancy word for "neighbors." These angles share a common vertex (the corner point) and a common arm (the line between them). Together, they form a perfect "L" shape.

• Example: The two angles created by the frame of a window.

B. Non-Adjacent Complementary Angles

These angles are "long-distance friends." They do not share a vertex or an arm. They might be on totally different parts of a page! However, as long as their numbers add up to 90°, they are still complementary.

• Example: An angle of 40° in one triangle and an angle of 50° in a completely different triangle.

Complementary Angles Examples

Let's look at some complementary angles examples to see how we can find a missing "best friend" angle.

Example 1: Finding the Missing Piece

If Angle X is 25°, what is its complementary angle?

• The Goal: The total must be 90°.

• The Math: $90 - 25 = 65$.

• Result: The complement of 25° is 65°.

Example 2: The Equal Pair

Is there a case where both complementary angles are exactly the same?

• The Math: We need a number that, when added to itself, equals 90. $45 + 45 = 90$.

• Result: A 45° angle is the complement of itself!

Example 3: Real-World Clock

Look at a clock. When it is 3:00, the hands make a 90° angle. If you imagine a line cutting through that space at the 1-minute mark (which is 6°), the two resulting angles (6° and 84°) are complementary!

Read More - Straight Angle (180°) – Definition, Degree, Properties, Examples

How to Identify Complementary Angles?

You can use the Complementary Angles Test:

1. Measure Angle 1.

2. Measure Angle 2.

3. Add them up.

4. Is the sum 90? If yes, they are complementary. If it is 89 or 91, they are NOT complementary!

Complementary Angles in Right-Angled Triangles

One of the most significant places to see complimentary angles in action is in a right-angled triangle. A triangle has three angles on the inside that always sum up to exactly $180^\circ$. If the triangle has a right angle, one of those angles is already $90^\circ$. There are still two angles left, and they can share exactly $90^\circ$.

The Interior Rule

The two non-right angles of a right-angled triangle (the acute angles) are always complimentary angles. For instance, if you have a triangle with one angle that is $90^\circ$ and one angle that is $40^\circ$, the third angle must be $50^\circ$ since $90 - 40 = 50$. They meet the notion of complementary angles precisely because 40 + 50 = 90.

Why does this matter?

• Trigonometry: This connection is what trigonometry is based on. Sine and Cosine are based on the ratios of these pairs that go together.

• Construction: Builders use this rule to ensure that roofs are pitched correctly and that stairs are set at a safe, consistent angle.

• Problem Solving: If you are given just one acute angle in a right triangle, you automatically know the other one thanks to the complementary angles definition.

Summary Table on Complementary angles

Let’s have a quick overlook about complementary angles. 

Read More - Right Angle Triangle: Definition, Properties, Formula & Examples

Difference Between Complementary Angles and Supplementary Angles

It is very easy to mix these two up!

• Complementary: Sum is 90° (Corner of a room).

• Supplementary: Sum is 180° (A straight line).

Curious Jr. Tip: Think of the alphabet! C comes before S. 90 comes before 180. So, Complementary is 90 and Supplementary is 180!

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