Solving For X: Complementary & Supplementary Angles

Nov 27, 2025 by Tom Lembong 52 views

Alright, guys, let's dive into the fascinating world of angles and algebra! We're going to tackle a problem where we need to find the value of 'x' given some information about angles. Specifically, we'll be working with complementary and supplementary angles. Buckle up, because it's going to be a fun ride!

Understanding Complementary and Supplementary Angles

First, let's quickly refresh our understanding of what complementary and supplementary angles are. These concepts are crucial for solving the problem at hand. Think of them as fundamental building blocks in our geometrical quest. Mastering these definitions will make the rest of the process much smoother and more intuitive.

Complementary Angles: These are two angles whose measures add up to 90 degrees. Imagine a right angle being split into two smaller angles; those two smaller angles are complementary. A classic example is a 30-degree angle and a 60-degree angle. When you add them together (30° + 60°), you get 90°, which confirms they are complementary. The key here is the sum – it always has to be exactly 90 degrees. Remembering this will help you quickly identify and work with complementary angles in various problems.

Supplementary Angles: These are two angles whose measures add up to 180 degrees. Picture a straight line; if you draw a ray from any point on that line, you create two angles that are supplementary. For instance, a 120-degree angle and a 60-degree angle are supplementary because 120° + 60° = 180°. Supplementary angles essentially form a straight line when placed adjacent to each other. Keeping this visual in mind can aid in recognizing and solving problems involving supplementary angles.

Understanding the difference between these two types of angles is super important. Complementary angles form a right angle (90°), while supplementary angles form a straight line (180°). Got it? Great! Now, let's move on to the problem.

Setting Up the Equation

Now that we've got our definitions sorted, let's tackle the algebraic expression we were given and translate that into something workable in the context of complementary and supplementary angles. This is where our algebra skills meet our geometry knowledge. It's like combining two superpowers to solve a challenging puzzle!

We have the expression: -20 + 3x - 40.

This expression represents an angle. The problem implies that this angle is related to either a complementary or supplementary angle. Since the problem statement doesn't explicitly state whether the angle is complementary or supplementary, we'll need to explore both possibilities to find the correct solution. This means we'll set up two different equations based on the two scenarios.

Scenario 1: Complementary Angle

If the angle is complementary, it means it adds up to 90 degrees with another angle. Let's assume the expression -20 + 3x - 40 represents one of the complementary angles. Then we can set up the equation:

(-20 + 3x - 40) + (another angle) = 90

However, we don't know what the "another angle" is. To simplify things, let's assume that -20 + 3x - 40 itself is the measure of an angle that is part of a complementary pair. In many problems, you'll be given more information to define the relationship, but in this case, we'll proceed by assuming the problem implies we are trying to find when this expression could represent an angle that could be part of a complementary or supplementary pair.
Scenario 2: Supplementary Angle

If the angle is supplementary, it means it adds up to 180 degrees with another angle. Similarly, we can set up the equation:

(-20 + 3x - 40) + (another angle) = 180

Again, we don't know what the "another angle" is, so we'll make a similar assumption as in Scenario 1. This allows us to solve for x under both conditions.

Let's simplify the expression first: -20 + 3x - 40 simplifies to 3x - 60. This simplified form will make our calculations easier in the next steps.

Solving for x: Complementary Case

Okay, let's assume that 3x - 60 is an angle, and we want to find out what value of x would make it a complementary angle. In other words, we want to find x such that 3x - 60 could be an angle that has a complement.

First, remember that for an angle to have a complement, it must be less than 90 degrees. Think about it: if an angle is 90 degrees or more, you can’t add another angle to it to get 90 degrees. So, we need to ensure that 3x - 60 < 90. This gives us an inequality to work with.

To solve this inequality, we'll follow these steps:

Add 60 to both sides of the inequality:

3x - 60 + 60 < 90 + 60

This simplifies to:

3x < 150
Divide both sides by 3:

3x / 3 < 150 / 3

This simplifies to:

x < 50

So, for the angle 3x - 60 to be a valid angle in a complementary relationship (i.e., for it to be less than 90 degrees), x must be less than 50.

Now, let's consider a scenario where 3x - 60 is one of the complementary angles, and we're trying to find x such that it satisfies the equation:

(3x - 60) + (complement) = 90

To make things simple, let's assume the 'complement' is also represented by the same expression (3x - 60). This isn't explicitly stated, but it allows us to solve for x directly:

(3x - 60) + (3x - 60) = 90

Combine like terms:

6x - 120 = 90

Add 120 to both sides:

6x = 210

Divide by 6:

x = 35

In this specific case, x equals 35. Let's plug this value back into our original expression to see if it makes sense:

3(35) - 60 = 105 - 60 = 45

So, one angle is 45 degrees. Since the angles are complementary and equal, the other angle is also 45 degrees (45 + 45 = 90). This confirms that when x is 35, the expression 3x - 60 can represent an angle in a complementary relationship.

Solving for x: Supplementary Case

Now, let's switch gears and consider the supplementary case. We're going to find the value of x that would make 3x - 60 a supplementary angle. Remember, supplementary angles add up to 180 degrees.

First, for 3x - 60 to be a valid angle in a supplementary relationship, it must be less than 180 degrees. If it were 180 degrees or more, you couldn't add another angle to it to get 180 degrees. So, we need to make sure that 3x - 60 < 180. Let's solve this inequality:

Add 60 to both sides:

3x - 60 + 60 < 180 + 60

This simplifies to:

3x < 240
Divide both sides by 3:

3x / 3 < 240 / 3

This simplifies to:

x < 80

So, for the angle 3x - 60 to be a valid angle in a supplementary relationship, x must be less than 80.

Now, let's find x such that the expression 3x - 60, when added to its supplement, equals 180:

(3x - 60) + (supplement) = 180

Again, to keep things simple, let's assume that the 'supplement' is also represented by the same expression (3x - 60). Solving for x with this assumption:

(3x - 60) + (3x - 60) = 180

Combine like terms:

6x - 120 = 180

Add 120 to both sides:

6x = 300

Divide by 6:

x = 50

In this specific case, x equals 50. Let's plug this value back into our expression to see if it makes sense:

3(50) - 60 = 150 - 60 = 90

So, one angle is 90 degrees. Since the angles are supplementary and equal, the other angle is also 90 degrees (90 + 90 = 180). This confirms that when x is 50, the expression 3x - 60 can represent an angle in a supplementary relationship.

Conclusion

Alright, guys, we've successfully navigated through the world of complementary and supplementary angles, and we've found the value of 'x' for both scenarios. Remember, understanding the basic definitions and then carefully setting up the equations is key to solving these types of problems. Keep practicing, and you'll become angle-solving pros in no time!

Key Takeaways:

Complementary angles add up to 90 degrees.
Supplementary angles add up to 180 degrees.
Carefully consider the conditions and assumptions when solving for variables in angle-related problems.

In summary:

For the complementary case (assuming the expression represents an angle in a complementary pair with itself), x = 35.
For the supplementary case (assuming the expression represents an angle in a supplementary pair with itself), x = 50.