Complementary Angles: Find X When M∠A=37° And M∠B=(3x+17)°

Understanding Complementary Angles

Let's dive into the fascinating world of geometry and tackle a common problem involving complementary angles. Complementary angles are a pair of angles that add up to 90 degrees. Think of them as two puzzle pieces that perfectly fit together to form a right angle. This fundamental concept is crucial in many areas of mathematics, from basic geometry to more advanced trigonometry. When we're given two angles that are complementary, we know a special relationship exists between them: their measures sum to exactly 90 degrees. This relationship allows us to set up equations and solve for unknown values, which is exactly what we'll be doing in this problem. Understanding this simple definition is the first step to unlocking the solution, and it's a concept that will serve you well as you continue your mathematical journey. Remember, complementary angles always equal 90 degrees when combined.

The Problem at Hand

In this specific problem, we are presented with two angles, Angle A and Angle B, which are stated to be complementary. This immediately tells us that their measures will sum to 90 degrees. We are given the measure of Angle A as 37 degrees ($m obreakspace extless extbraceleft extgreater extbraceright extbraceleft m extbraceleft extgreater extbraceleft extgreater A=37^{\circ}$). For Angle B, we have an expression involving an unknown variable, x: its measure is $(3x + 17)$ degrees ($m obreakspace extless extbraceleft extgreater extbraceright extbraceleft m extbraceleft extgreater extbraceleft extgreater B=(3x+17)^{\circ}$). Our goal is to find the value of x that satisfies this complementary relationship. This means we need to use the information given to create an equation that we can solve for x. The beauty of geometry problems like this is how they connect seemingly separate pieces of information into a cohesive whole, leading us to a definitive answer. The setup is straightforward: we have a known angle and an angle expressed with a variable, and the constraint that they are complementary. This is the perfect recipe for an algebraic solution within a geometric context.

Setting Up the Equation

Now that we understand the definition of complementary angles and the specifics of our problem, we can set up the equation. Since Angles A and B are complementary, their measures must add up to 90 degrees. We can write this relationship as an equation:

$$m \angle A + m \angle B = 90^{\circ}$$

Substitute the given values into this equation:

$$37^{\circ} + (3x + 17)^{\circ} = 90^{\circ}$$

It's important to note that we can drop the degree symbols once we've established the equation, as we are solving for a numerical value of 'x'. So, the equation becomes:

$$37 + (3x + 17) = 90$$

This equation now represents the core of our problem. We have successfully translated the geometric concept of complementary angles into an algebraic expression. The next steps will involve simplifying and solving this equation to isolate the variable 'x'. This process highlights the powerful synergy between algebra and geometry, where abstract shapes and relationships can be described and manipulated using numerical and symbolic tools. By setting up this equation correctly, we've laid the groundwork for finding the precise value of 'x' that makes Angle B measure what it needs to be for the two angles to be complementary.

Solving for x

With our equation $37 + (3x + 17) = 90$ established, we can now proceed to solve for x. Our primary objective is to isolate 'x' on one side of the equation. First, let's combine the constant terms on the left side of the equation:

$$37 + 17 = 54$$

So, the equation simplifies to:

$$54 + 3x = 90$$

Next, we want to get the term with 'x' by itself. To do this, subtract 54 from both sides of the equation:

$$3x = 90 - 54$$

$$3x = 36$$

Now, to find the value of 'x', divide both sides of the equation by 3:

$$x = \frac{36}{3}$$

$$x = 12$$

Therefore, the value of x is 12. This step-by-step algebraic manipulation is key to solving for the unknown. We've systematically removed the known values from the 'x' term, revealing its solitary value. This process is fundamental to algebra and is directly applied here to solve our geometric problem. The result, x=12, is the numerical answer we were seeking. It's a clean, integer value, which often indicates that the problem was designed to have a straightforward solution. This is the power of algebra in action, turning a geometric relationship into a solvable equation.

Verification of the Solution

To ensure our answer is correct, it's always a good practice to verify the solution. We found that $x=12$. Let's substitute this value back into the expression for Angle B:

$$m \angle B = (3x + 17)^{\circ}$$

$$m \angle B = (3(12) + 17)^{\circ}$$

$$m \angle B = (36 + 17)^{\circ}$$

$$m \angle B = 53^{\circ}$$

Now, let's check if Angle A and Angle B are indeed complementary by adding their measures:

$$m \angle A + m \angle B = 37^{\circ} + 53^{\circ}$$

$$m \angle A + m \angle B = 90^{\circ}$$

Since the sum of the measures of Angle A and Angle B is 90 degrees, our value of $x=12$ is correct. This verification step is crucial in confirming that our calculations were accurate and that the value of 'x' we found indeed satisfies the condition of complementary angles. It's a satisfying moment when the numbers align perfectly, validating the entire process from understanding the concept to the final algebraic solution. This confirmation builds confidence in our mathematical abilities.

Conclusion and Further Exploration

In conclusion, we have successfully determined that the value of x is 12 by utilizing the property of complementary angles. This problem demonstrated how a fundamental geometric definition can be translated into an algebraic equation, which can then be solved using basic algebraic techniques. Understanding complementary angles and how to set up and solve equations is a cornerstone of geometry and a valuable skill in mathematics. Whether you're a student learning these concepts for the first time or looking for a refresher, remember that practice is key! Keep exploring different types of angle relationships, such as supplementary angles (which add up to 180 degrees) and vertical angles. Each type of relationship offers unique properties that can be used to solve a wide variety of problems.

For those interested in delving deeper into the world of geometry and angles, I highly recommend exploring resources from trusted educational websites. You can find extensive information and practice problems on Khan Academy, which offers a comprehensive approach to learning geometry and other mathematical subjects. Their interactive lessons and exercises are excellent for reinforcing your understanding and building your problem-solving skills. Keep exploring, keep learning, and enjoy the process of discovering the elegant logic of mathematics!