**Introduction**

Angles are fundamental to our understanding of the world around us. Whether it is the corner of a book, the ramp’s incline, or the sunflower’s tilt toward the sun, angles are everywhere! This article focuses on one particular pair of angles known as “complementary angles.” Grab your protractor, and let us dive in!

**Grade Appropriateness**

Complementary angles are typically introduced between the 4th and 7th grades. It means if you are between 8 and 15 years old, you are at the right age to get to know them better!

**Math Domain**

Complementary angles fall under the “Geometry” domain of mathematics. This domain explores shapes, sizes, space properties, and angles in nature.

**Applicable Common Core Standards**

*4.MD.C.7:* Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts.

*7.G.B.5:* Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

**Definition of the Topic**

Two angles are said to be complementary if their measures sum up to 90 degrees. If you place them together, they will form a right angle, which looks like the corner of a square or rectangle.

**Key Concepts**

** Right Angle:** This is an angle that measures exactly 90 degrees.

** Angle Measure:** The size of an angle, measured in degrees.

** Complementary Angles:** These are two angles whose measures add up to 90 degrees.

**Discussion with Illustrative Examples**

An **angle** in geometry is the figure created when two rays intersect at a common point known as a vertex. An angle measures the amount of turn of its rays in **degrees**.

**Types of Angles**

A **zero degree angle** has angle rays that overlap. In the figure below, ∠LMN shows zero degrees in measure.

An **acute angle** measures less than 90° or between 0 and 90 degrees. Here, ∠b illustrates an acute angle.

A **right angle** has a measure that is exactly 90°. The image below shows a right angle.

An **obtuse angle** measures greater than 90° but less than 180°. In the image below, ∠b shows an obtuse angle.

A **straight angle** measures exactly 180°. The figure below illustrates a straight angle.

A **reflex angle** measures greater than 180° but less than 360°. The image below shows ∠b, which is a reflex angle.

A **full rotation angle** is an angle that measures exactly 360°. A full rotation angle is formed when one ray makes a complete rotation to form an angle.

**Complementary Angles**

Two angles are said to be complementary if their measures sum up to 90 degrees.

Complementary angles are only limited to two angles; hence, three or more angles with a sum of 90° cannot be complementary.

For example, 50° is the complement of 40° while 60° is the complement of 30°

Complementary angles can be either adjacent or non-adjacent.

**Adjacent Complementary Angles**

Adjacent complementary angles are angles with a common vertex and side (arm). In the figure below, ∠LMN and ∠NMO are adjacent angles with a common vertex and side. Also, they have a sum of 90°. Hence, these two angles are adjacent complementary angles.

**Non-Adjacent Complementary Angles**

Angles are non-adjacent angles as they NEITHER have a common vertex nor a common side. In the figure below, ∠LMN and ∠XYZ add up to 90° and do not have a common vertex and side. Therefore, these two angles are non-adjacent complementary angles.

**Examples with Solution**

**Example 1**

Find the complement of a 25-degree angle.

**Solution**

90°-25°=65°

Thus, 65° is the complement of 25°.

**Example 2**

An angle measures 32°. Find its complementary angle.

**Solution**

90°-32°=58°

Hence, 58° is the complement of 32°.

**Example 3**

If m∠RQS=70°, find m∠PQR.

**Solution**

Since m∠PQS=90°, so we have,

m∠PQS-m∠RQS=m∠PQR

90°-70°=20°

Thus, m∠PQR=20°.

**Example 4**

The angles below are complementary. Find the value of x.

**Solution**

Since ∠ABC and ∠PQR are complementary angles, we have,

m∠ABC+m∠PQR=90°

48°+5x+2°=90°

5x+2°=90°-48°

5x+2°=42°

5x=42°-2°

5x=40°

5×5=40°5

x=8°

From the solution, the complement of ∠ABC is 42°.

Calculating the value of x gives us 8°.

**Real-life Application with Solution**

Suppose you are setting up a ramp for skateboarding. You want the bottom of the ramp to have an incline of 20 degrees. To ensure safety, you want to find the angle at which the top of the ramp will meet the ground (it should be a right angle with the incline).

**Solution**

Complementary angle to 20°=90°-20°=70°

So, the top of the ramp will meet the ground at an angle of 70 degrees.

**Practice Test**

1. What is the complementary angle of a 15-degree angle?

2. An angle measures 45 degrees. What is its complement?

3. If one angle is twice the measure of its complement, what are the angles?

4. An angle measures x degrees. Write an expression for its complementary angle.

5. (Word problem) Lisa cut a piece of pie at an angle of 35 degrees. What angle is left to make it a right-angle slice?

6. (Word problem) A ladder is leaning against a wall creating an angle of 75 degrees with the ground. What angle does it make with the wall?

*Answers:*

1. 75°is the complement of 15°.

**2. **The complement of an angle that measures 45°is 45°.

3. One angle measure 60° while 30°is its complement. 60° is twice the measure of 30° and their sum is 90°.

4. If an angle measures x degrees, its complement must be 90°-x degrees.

5. 90°-35°=55°

6. 90°-75°=15°

**Frequently Asked Questions (FAQs)**

**Are complementary angles always adjacent?**

No. While angles can be adjacent, it is not a requirement for complementary angles to be adjacent.

**Can one angle be bigger than the other in a complementary pair?**

Absolutely! If their sum is 90 degrees, they are complementary.

**What differentiates complementary from supplementary angles?**

Supplementary angles add up to 180 degrees, whereas complementary angles add up to 90 degrees.

**Do complementary angles always form a right angle when combined?**

Yes! When you put them together, they will always make a right angle.

**If I know the measure of one angle, how can I find its complement?**

Subtract the given angle’s measure from 90 degrees to find its complement.

Now, young mathematicians, you’re equipped with all you need to understand and identify complementary angles. Look around you; they are everywhere! In architecture or nature, complementary angles are crucial in shaping our world.

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