**Table of Contents**

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**What are Complementary Angles in Geometry?**

**Definition of Complementary Angles**

**Two angles are said to be complementary angles if they add up to 90 degrees.**

In other words, when complementary angles are put together, they form a right angle (90 degrees).

\(\angle 1\) and \(\angle 2\) are complementary if |

In this case, \(\angle 1\) and \(\angle 2\) are called **complements** of each other.

**Examples of Complementary Angles**

Here, \(60^\circ+30^\circ = 90^\circ\)

Hence, from the "Definition of Complementary Angles", these two angles are complementary.

Each angle among the complementary angles is called the "complement" of the other angle.

Here,

- 60
^{o}is the complement of 30^{o} - 30
^{o}is the complement of 60^{o}

Some real-life examples of complementary angles are as follows:

The two angles in each of the above figures are adjacent (it means that they have a common vertex and a common arm).

But, do complementary angles always need to be adjacent?

No, they can be non-adjacent as well.

Let us see how.

**Adjacent and Non-Adjacent Complementary Angles (With Illustrations)**

There are two types of complementary angles in geometry.

- Adjacent Complementary Angles
- Non-adjacent Complementary Angles

**Adjacent Complementary Angles**

Two complementary angles with a common vertex and a common arm are called **adjacent complementary angles**.

**Example**

Here, \(\angle COB\) and \(\angle AOB\) are adjacent angles as they have a common vertex \(O\) and a common arm \(OB\).

They also add up to 90 degrees. i.e.,

\[\angle COB + \angle AOB = 70^\circ+20^\circ=90^\circ\]

Thus, these two angles are adjacent complementary angles.

You can observe adjacent complementary angles using the following illustration.

Move point C to change the angles and click "GO".

**Non-Adjacent Complementary Angles**

Two complementary angles that are NOT adjacent are said to be **non-adjacent complementary angles**.

**Example**

Here, \(\angle ABC\) and \(\angle PQR\) are non-adjacent angles as they neither have a common vertex nor a common arm.

Also, they add up to 90 degrees. i.e.,

\[\angle ABC+ \angle PQR = 50^\circ+40^\circ=90^\circ\]

Thus, these two angles are non-adjacent complementary angles.

When non-adjacent complementary angles are put together, they form a right angle.

You can observe this using the following illustration.

Move the first slider to change the angles and move the second slider to see how the angles are complementary.

**How to Find Complement of an Angle? (With an Activity)**

We know that the sum of two complementary angles is 90 degrees and each of them is said to be a "complement" of each other.

Thus, the complement of an angle is found by subtracting it from 90 degrees.

Complement of \(x^\circ\) is \((90-x)^\circ\) |

**Example**

What is the complement of 57^{o}?

**Solution**

The complement of 57^{o} is obtained by subtracting it from 90^{o}

Thus, its complement is:

\[(90-57)^\circ =33^\circ\]

Here is an activity on "Finding the Complement of an Angle".

You can click and drag the "Orange" dot to change the angles and then you can enter the complement of the given angle.

It will then indicate whether your answer is correct.

**Differences between Complementary and Supplementary Angles**

Complementary |
Supplementary Angles |
---|---|

Two angles are said to be complementary if they add up to 90 degrees. |
Two angles are said to be supplementary if they add up to 180 degrees. |

Complement of an angle \(x^\circ\) is \((90-x)^\circ\) |
Supplement of an angle \(x^\circ\) is \((180-x)^\circ\) |

- "S" is for "Supplementary" and "S" is for "Straight".

Hence, you can remember that two "Supplementary" angles when put together form a "Straight" angle. - "C" is for "Complementary" and "C" is for "Corner".

Hence, you can remember that two "Complementary" angles when put together form a "Corner (right)" angle.

**Complementary Angle Theorem (with Illustration)**

**The complementary angle theorem states, "If two angles are complementary to the same angle, then they are congruent to each other".**

**Proof of Complementary Angle Theorem**

Consider the following figure:

Let us assume that \(\angle POQ\) is complementary to \(\angle AOP\) and \(\angle BOQ\).

By the definition of complementary angles,

\[ \begin{align} \angle POQ + \angle AOP &= 90^\circ\\[0.3cm] \angle POQ + \angle BOQ &=90^\circ \end{align}\] From the above two equations, we can say that \[\angle POQ + \angle AOP=\angle POQ + \angle BOQ\] Subtracting \(\angle POQ \) from both sides, \[\angle AOP = \angle BOQ\] Hence, the theorem is proved.

You can visualize the Complementary Angle Theorem using the following illustration.

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**Solved Examples**

Example 1 |

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

**Solution:**

In the given figure, \(x\) and 62^{o} are complementary as they form a right angle.

Hence, their sum is 90^{o}

\[ \begin{align} x +62^\circ &= 90^\circ \\[0.2cm] x &= 90^\circ-62^\circ\\[0.2cm] x&= 28^\circ \end{align}\]

\(\therefore\) \(x=28^\circ\) |

Example 2 |

Find the values of \(\angle A\) and \(\angle B\) if \(\angle A\) and \(\angle B\) are complementary such that \(\angle A=3x-25\) and \(\angle B=6x−65\).

**Solution:**

Since \(\angle A\) and \(\angle B\) are complementary, their sum is 90^{o}.

\[ \begin{align} \angle A+\angle B &=90\\[0.2cm] (3x-25)+(6x-65)&=90\\[0.2cm] 9x - 90&=90\\[0.2cm] 9x&=180\\ x &= 20 \end{align}\]

Therefore, \[ \begin{align} \angle A &= 3(20)-25 = 35^\circ\\[0.2cm] \angle B &= 6(20)-65 =55^\circ \end{align} \]

\(\therefore\) \( \begin{align} \angle A &= 35^\circ\\[0.2cm] \angle B & =55^\circ \end{align} \) |

Example 3 |

Find the value of \(x\) if the following two angles are complementary.

**Solution:**

Since the given two angles are complementary, their sum is 90^{o}.

\[ \begin{align} \dfrac{x}{2}+ \dfrac{x}{3}&=90\\[0.2cm] \dfrac{5x}{6}&=90\\[0.2cm] x&=90 \times \dfrac{6}{5}\\[0.2cm] x &= 108\end{align}\]

\(\therefore\) \(x = 108\) |

Example 4 |

Two angles are complementary.

The measure of the larger angle is 5 degrees more than 4 times the measure of the smaller angle.

What is the measure of the larger angle in degrees?

**Solution:**

Let us assume that the two complementary angles are \(x\) (larger) and \(y\) (smaller).

By the given information,\[x = 4y+5\]

Since the two angles are complementary, their sum is 90^{o}.

\[ \begin{align} x+y&=90\\[0.2cm] (4y+5)+y &=90& [\because x=4y+5]\\[0.2cm] 5y+5&=90\\[0.2cm] 5y&=85\\[0.2cm] y&=17\end{align} \]

Thus, the larger angle is, \[x = 4(17)+5=73^\circ\]

\(\therefore\) Larger angle = \(73^\circ\) |

- In the given figure, PQ is perpendicular to PR and ∠QPS=56°. Find ∠RPT.

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**Practice Questions**

**Maths Olympiad Sample Papers**

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

- IMO Sample Paper Class 1
- IMO Sample Paper Class 2
- IMO Sample Paper Class 3
- IMO Sample Paper Class 4
- IMO Sample Paper Class 5
- IMO Sample Paper Class 6
- IMO Sample Paper Class 7
- IMO Sample Paper Class 8
- IMO Sample Paper Class 9
- IMO Sample Paper Class 10

To know more about the Maths Olympiad you can **click here**

**Frequently Asked Questions (FAQs)**

## 1. How do you find a complementary angle?

If the sum of two angles is 90 degrees, then we say that they are complementary.

Thus, the complement of an angle is obtained by subtracting it from 90.

For example, the complement of \(40^\circ\) is \(90-40=50^\circ\).

## 2. What is the complementary angle of 40 degrees?

The complement of an angle is obtained by subtracting it from 90 degrees.

Thus, the complement of \(40^\circ\) is \(90-40=50^\circ\).

## 3. How do you find the value of x in complementary angles?

If two angles in terms of \(x\) are given to be complementary, we just set their sum equal to 90 degrees and solve the resultant equation.

You can refer to "Example 2" and "Example 3" under the "Solved Examples" section of this page.

## 4. How do you find the ratio of two complementary angles?

The ratio of two complementary angles is found just like how we find the ratio of any two quantities.

For example, the ratio of the complementary angles \(40^\circ\) and \(50^\circ\) is \(\dfrac{40}{50}= \dfrac{4}{5}\)