What are Supplementary and Complementary Angles?

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Supplementary and Complementary Angles – Introduction

In geometry, understanding angle relationships is essential, especially when we talk about supplementary and complementary angles. These concepts are key tools for solving puzzles about missing angles without even using a protractor. Mastering these angles helps you tackle various geometric problems more efficiently and see the shapes in the world with a new perspective. Whether you're working on a school project, designing something creative, or just curious about how things fit together, knowing about these angles is incredibly useful. It’s all about applying these ideas to figure out angles in real-life scenarios, boosting your skills in geometry and beyond.

Definition Image Examples
Complementary Angles Two angles whose sum is 90 degrees. Two angles measuring 30 and 60 degrees.
Supplementary Angles Two angles whose sum is 180 degrees. Two angles measuring 110 and 70 degrees.

Having a basic understanding of angles and measuring angles will be helpful when it comes to understanding angle pairs.

Complementary Angles

Complementary angles are two angles whose measures have a sum of exactly 90 degrees. When combined, they form a right angle.

Identifying Complementary Angles

To determine if two angles are complementary, you can calculate the sum to find out if it is equal to $90^\circ$.

Are the angles $40^\circ$ and $45^\circ$ complementary?

• Add the angles together $40+45$.
• Answer the question, is this sum exactly $90^\circ$? If you answered yes, they are complementary! If the answer is no, as in $40+45=85$, then the answer is no, these angles are not complementary.

Are the angles $53^\circ$ and $37^\circ$ complementary?

• Add the angles together $53+37$.
• These angles have a sum of 90 degrees since $53+37=90^\circ$. These two angles form a right angle.
Are the following angles complementary? $30^\circ$ and $60^\circ$
Are the following angles complementary? $45^\circ$ and $55^\circ$
Are the following angles complementary? $80^\circ$ and $10^\circ$

Finding a Complementary Angle

If one of the two angles are known, using subtraction can help to find the angle that is complementary to the given angle.

What is the measure of the missing complementary angle?

• Since a right angle is equal to 90 degrees, this is our total.
• The known value, in this case, $41^\circ$ can be subtracted from the total to find the missing complementary angle. $90-41=49$.
• $41^\circ$ and $49^\circ$ are complementary angles because they have a sum of $90^\circ$.
If one angle measures $67^\circ$, what is the measure of the angle complementary to it?
If one angle measures $15^\circ$, what is the measure of the angle complementary to it?
If one angle measures $50^\circ$, what is the measure of the angle complementary to it?

Finding missing angles is a skill that will help solve problems involving more complex angle relationships.

Supplementary Angles

Supplementary angles are two angles whose measures sum to 180 degrees, together forming a straight line.

Think of a straight ruler as an analogy. If two angles were placed at one end of the ruler, with their vertexes meeting at the same point and their sides extending along the ruler's length, these angles would be supplementary if they covered the entire 180-degree space opposite each other.

Identifying Supplementary Angles

To determine if two angles are supplementary, you can calculate the sum to find out if it equals $180^\circ$.

Are the angles $120^\circ$ and $60^\circ$ supplementary?

• Add the angles together $120+60$.
• Answer the question, is this sum exactly $180^\circ$? If you answered yes, they are supplementary!

Are the angles $130^\circ$ and $40^\circ$ supplementary?

• Add the angles together $130+40$.
• These angles have a sum of 170 degrees since $130+40=170^\circ$. These two angles do not form a straight line.
Are the following angles supplementary? $150^\circ$ and $30^\circ$
Are the following angles supplementary? $100^\circ$ and $80^\circ$
Are the following angles supplementary? $90^\circ$ and $95^\circ$

Finding a Supplementary Angle

If one of two angles is known, using subtraction can help to find the angle that is supplementary to the given angle.

What is the measure of the missing supplementary angle?

• Since a straight line is equal to 180 degrees, this is our total.
• The known value, in this case, $38^\circ$, can be subtracted from the total to find the missing supplementary angle. $180-38=142$.
• $38^\circ$ and $142^\circ$ are supplementary angles because they have a sum of $180^\circ$.
If one angle measures $110^\circ$, what is the measure of the angle supplementary to it?
If one angle measures $45^\circ$, what is the measure of the angle supplementary to it?
If one angle measures $75^\circ$, what is the measure of the angle supplementary to it?

Supplementary and Complementary Angles – Application

A skateboard ramp forms an angle of $140^\circ$ with the ground. What is the measure of the angle between the ramp and a vertical line from the ground?

• Since a straight line measures $180^\circ$, and the ramp forms a $140^\circ$ angle with the ground, we're looking for the supplementary angle.
• Subtract the ramp's angle from $180^\circ$ to find the missing angle: $180 - 140 = 40$.
• The angle between the ramp and a vertical line from the ground is $40^\circ$.

A classroom door is opened making a $30^\circ$ angle with the door frame. What is the measure of the angle formed between the door and the wall it's attached to?
During a game, a basketball player aims to make a shot by throwing the ball at an angle. If the angle of elevation from the player to the hoop is $55^\circ$, what is the complementary angle to this angle of elevation?
In a digital art project, you're designing a scene where two spotlights cross paths on a wall. If one spotlight creates a $120^\circ$ angle with the wall, what is the angle created by the other spotlight to make them supplementary?

Supplementary and Complementary Angles – Summary

Key Learnings from this Text:

• Complementary angles sum to 90 degrees and form a right angle.
• Supplementary angles sum to 180 degrees and create a straight line.
• Understanding these angle pairs is essential for solving various geometric problems without the use of a protractor.

Supplementary and Complementary Angles – Frequently Asked Questions

Can two acute angles be supplementary?
Can an obtuse angle be part of a complementary pair?
What if I only know one angle of a supplementary pair?
How do complementary and supplementary angles relate to right angles and straight lines?
Can angles be both complementary and supplementary?
How do you find the measure of a missing angle in a right triangle?
Why is it impossible for a pair of angles to be complementary if one is a reflex angle?
Do supplementary angles always create a straight angle?

TranscriptWhat are Supplementary and Complementary Angles?

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What are Supplementary and Complementary Angles? exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video What are Supplementary and Complementary Angles?.
• What are complementary and supplementary angles?

Hints

There are different ways to help you remember the difference between complementary and supplementary angles.

Remember that complementary angles add up to 90 degrees, just like the corners of a square or rectangle. Think of "C" for "corner" or "right angle."

Supplementary angles add up to 180 degrees, forming a straight line when adjacent. Associate "S" with "straight line."

Solution

When two angles share a side and a vertex, they are known as adjacent angles. When two adjacent angles have a sum of $90^\circ$ they are called complementary angles. Angles that have a measurement of $90$ degrees are also known as right angles.

When adjacent angles are on a straight line, they have a measurement of $\bf{180^\circ}$. These angle pairs are known as supplementary angles.

• Understand the difference between Complementary and Supplementary Angles.

Hints

Complementary angles are two angles whose measures add up to 90 degrees, often forming a right angle when adjacent.

Supplementary angles are two angles whose measures add up to 180 degrees, often forming a straight line when adjacent.

Solution

Complementary Angles (Sum of $90^\circ$)

• $82^\circ$ and $8^\circ$
• $73^\circ$ and $17^\circ$
• $45^\circ$ and $45^\circ$
• $50^\circ$ and $40^\circ$
• $15^\circ$ and $75^\circ$
Supplementary Angles (Sum of $180^\circ$)

• $57^\circ$ and $123^\circ$
• $150^\circ$ and $30^\circ$
• $101^\circ$ and $79^\circ$
• $99^\circ$ and $81^\circ$
• $129^\circ$ and $51^\circ$
• Apply your knowledge of complementary angles.

Hints

Complementary angles are two angles that have a sum of $90$ degrees.

The square in the corner of an angle indicates that it is a right angle with a measurement of 90 degrees.

Solution

To find the solution you can subtract the $62$ from $90$ because these angles are complementary and have a sum of 90 degrees.

$90-62=28$

$x=28^\circ$

• Apply your knowledge of supplementary angles.

Hints

Supplementary angles span the length of a straight line. Remember that two angles that span a line add up to 180 degrees.

Look at this example to better visualize supplementary angles.

Solution

These two angles have a sum of $180$ degrees and are called supplementary angles. To find the missing angle, $121$ can be subtracted from $180$.

$180-121=59$

$y=59^\circ$

• What are the definitions of different types of angles?

Hints

Adjacent means 'next to.' In geometry, adjacent angles are like two pizza slices touching at a corner, sharing one side. They start from the same point and lie beside each other.

Straight lines have an angle measurement of 180 degrees.

What type of angles add to 180 degrees?

The sum of two supplementary angles are always greater than the sum of two complementary angles.

When two angles have a sum of 90 degrees, they also form what's known as a right angle.

Solution

Straight Angle - Angles with a measurement of 180 degrees.

Adjacent Angles - Angles sharing a common side and vertex.

Complementary Angles - Two angles that add to 90 degrees.

Supplementary Angles - Two angles that add to 180 degrees.

• Apply your understanding of complementary and supplementary angles.

Hints

The symbol $\angle$ means angle.

Instead of saying "angle x", it can be written $\angle x$.

Complementary angles are two angles whose measures add up to $90^\circ$. They often form a right angle when adjacent. For example, if one angle measures $30^\circ$, its complementary angle must measure $60^\circ$, because $30^\circ + 60^\circ = 90^\circ$.

Supplementary angles are two angles whose measures add up to $180^\circ$. They typically form a straight line when adjacent. For instance, if one angle measures $110^\circ$, its supplementary angle must measure $70^\circ$, as $110^\circ + 70^\circ = 180^\circ$.

There are a total of 3 correct answers for this problem.

Solution

True statements:

• $\angle a = 49^\circ$
• $\angle b = 49^\circ$
• $\angle a = \angle b$