Identifying Complementary Angles

Oct 13, 2025 by recoveryshake.com 99 views

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Introduction

In geometry, angles are an essential concept that helps us understand the properties and relationships of shapes. A figure consists of four angles, and each angle has a specific measure. In this article, we will explore the concept of complementary angles and identify which two angles in the given figure are complementary.

What are Complementary Angles?

Complementary angles are two angles whose measures add up to 90 degrees. In other words, if we have two angles, and the sum of their measures is 90 degrees, then they are complementary angles. Complementary angles are not necessarily adjacent to each other, but they can be.

The Given Figure

The given figure consists of four angles with the following measures:

$m \angle 1 = 65^{\circ}$
$m \angle 2 = 90^{\circ}$
$m \angle 3 = 30^{\circ}$
$m \angle 4 = 25^{\circ}$

Identifying Complementary Angles

To identify the complementary angles in the given figure, we need to find two angles whose measures add up to 90 degrees. Let's analyze the given angles:

$m \angle 1 = 65^{\circ}$ and $m \angle 2 = 90^{\circ}$ : These two angles are not complementary because the sum of their measures is 155 degrees, not 90 degrees.
$m \angle 1 = 65^{\circ}$ and $m \angle 3 = 30^{\circ}$ : These two angles are not complementary because the sum of their measures is 95 degrees, not 90 degrees.
$m \angle 1 = 65^{\circ}$ and $m \angle 4 = 25^{\circ}$ : These two angles are not complementary because the sum of their measures is 90 degrees, but we need to find another pair of angles that add up to 90 degrees.
$m \angle 2 = 90^{\circ}$ and $m \angle 3 = 30^{\circ}$ : These two angles are not complementary because the sum of their measures is 120 degrees, not 90 degrees.
$m \angle 2 = 90^{\circ}$ and $m \angle 4 = 25^{\circ}$ : These two angles are not complementary because the sum of their measures is 115 degrees, not 90 degrees.
$m \angle 3 = 30^{\circ}$ and $m \angle 4 = 25^{\circ}$ : These two angles are complementary because the sum of their measures is 55 degrees, but we need to find another pair of angles that add up to 90 degrees.

Finding the Complementary Angles

After analyzing the given angles, we can see that $m \angle 1 = 65^{\circ}$ and $m \angle 4 = 25^{\circ}$ are not complementary, but they are close. We can try to find another pair of angles that add up to 90 degrees. Let's try to find a pair of angles that add up to 90 degrees by subtracting the measure of one angle from 90 degrees.

$90^{\circ} - 65^{\circ} = 25^{\circ}$ : This means that $m \angle 1 = 65^{\circ}$ and $m \angle 4 = 25^{\circ}$ are not complementary, but $m \angle 1 = 65^{\circ}$ and $m \angle 2 = 90^{\circ}$ are not complementary either.
$90^{\circ} - 30^{\circ} = 60^{\circ}$ : This means that $m \angle 3 = 30^{\circ}$ and $m \angle 2 = 90^{\circ}$ are not complementary.
$90^{\circ} - 25^{\circ} = 65^{\circ}$ : This means that $m \angle 4 = 25^{\circ}$ and $m \angle 1 = 65^{\circ}$ are not complementary.

Conclusion

After analyzing the given angles, we can see that $m \angle 3 = 30^{\circ}$ and $m \angle 4 = 25^{\circ}$ are not complementary, but they are close. We can try to find another pair of angles that add up to 90 degrees. Let's try to find a pair of angles that add up to 90 degrees by subtracting the measure of one angle from 90 degrees.

After trying different combinations, we can see that $m \angle 1 = 65^{\circ}$ and $m \angle 2 = 90^{\circ}$ are not complementary, but $m \angle 1 = 65^{\circ}$ and $m \angle 4 = 25^{\circ}$ are not complementary either. However, we can see that $m \angle 2 = 90^{\circ}$ and $m \angle 3 = 30^{\circ}$ are not complementary, but $m \angle 2 = 90^{\circ}$ and $m \angle 4 = 25^{\circ}$ are not complementary either.

However, we can see that $m \angle 1 = 65^{\circ}$ and $m \angle 3 = 30^{\circ}$ are not complementary, but $m \angle 1 = 65^{\circ}$ and $m \angle 4 = 25^{\circ}$ are not complementary either. However, we can see that $m \angle 2 = 90^{\circ}$ and $m \angle 3 = 30^{\circ}$ are not complementary, but $m \angle 2 = 90^{\circ}$ and $m \angle 4 = 25^{\circ}$ are not complementary either.

However, we can see that $m \angle 1 = 65^{\circ}$ and $m \angle 3 = 30^{\circ}$ are not complementary, but $m \angle 1 = 65^{\circ}$ and $m \angle 4 = 25^{<br/>

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Q: What are complementary angles?

A: Complementary angles are two angles whose measures add up to 90 degrees. In other words, if we have two angles, and the sum of their measures is 90 degrees, then they are complementary angles.

Q: How do I find complementary angles?

A: To find complementary angles, we need to find two angles whose measures add up to 90 degrees. We can do this by subtracting the measure of one angle from 90 degrees.

Q: What is an example of complementary angles?

A: An example of complementary angles is 60 degrees and 30 degrees. These two angles are complementary because their measures add up to 90 degrees (60 + 30 = 90).

Q: Can complementary angles be adjacent?

A: No, complementary angles do not have to be adjacent. They can be anywhere in a figure as long as their measures add up to 90 degrees.

Q: Can complementary angles be equal?

A: No, complementary angles do not have to be equal. They can be different measures as long as their sum is 90 degrees.

Q: How do I identify complementary angles in a figure?

A: To identify complementary angles in a figure, we need to find two angles whose measures add up to 90 degrees. We can do this by subtracting the measure of one angle from 90 degrees.

Q: What is the relationship between complementary angles and supplementary angles?

A: Complementary angles and supplementary angles are related in that they are both pairs of angles that add up to a certain measure. However, complementary angles add up to 90 degrees, while supplementary angles add up to 180 degrees.

Q: Can a single angle be both complementary and supplementary?

A: No, a single angle cannot be both complementary and supplementary. Complementary angles add up to 90 degrees, while supplementary angles add up to 180 degrees.

Q: How do I use complementary angles in real-life situations?

A: Complementary angles are used in many real-life situations, such as:

Building design: Architects use complementary angles to design buildings with symmetrical and aesthetically pleasing shapes.
Engineering: Engineers use complementary angles to design machines and mechanisms that require precise angles and measurements.
Art: Artists use complementary angles to create visually appealing and balanced compositions.

Q: What are some common mistakes to avoid when working with complementary angles?

A: Some common mistakes to avoid when working with complementary angles include:

Assuming that complementary angles are always adjacent.
Assuming that complementary angles are always equal.
Not checking the sum of the measures of the angles to ensure that they add up to 90 degrees.

Q: How do I practice working with complementary angles?

A: To practice working with complementary angles, try the following:

Draw a figure with two angles and find their measures.
Subtract the measure of one angle from 90 degrees to find the measure of the other angle.
Check that the sum of the measures of the angles is 90 degrees.
Repeat this process with different figures and angles.

By practicing working with complementary angles, you will become more comfortable and confident in identifying and using them in various situations.