Supplementary Angles Problem Solving Find Measure Of Angle X

In the realm of geometry, angles play a fundamental role, and understanding their relationships is crucial for solving various problems. Among these relationships, the concept of supplementary angles stands out as a cornerstone. Supplementary angles are two angles whose measures add up to 180 degrees. This fundamental definition forms the basis for a myriad of geometric deductions and calculations. In this article, we will delve into a specific problem involving supplementary angles, where the measure of one angle is expressed in terms of the other. By applying algebraic principles and a clear understanding of the supplementary angle relationship, we will systematically unravel the solution.

This problem serves as an excellent example of how mathematical concepts intertwine to create a solvable puzzle. It highlights the importance of not only memorizing definitions but also grasping the underlying principles and applying them creatively. By dissecting the problem statement, setting up equations, and employing algebraic techniques, we will arrive at the measure of the unknown angle. This process not only reinforces our understanding of supplementary angles but also hones our problem-solving skills in mathematics. So, let's embark on this journey of geometric exploration and discover the measure of angle X in this intriguing scenario.

The problem presents us with a scenario involving two angles, X and Y, which are described as supplementary. This immediately tells us that the sum of their measures is 180 degrees. However, the problem doesn't stop there; it introduces an additional piece of information: angle X is 3 times the measure of angle Y. This is a crucial element that allows us to establish a relationship between the two angles and ultimately solve for their individual measures. This type of problem effectively bridges the gap between geometry and algebra, requiring us to translate geometric relationships into algebraic equations.

To effectively tackle this problem, we need to carefully dissect the given information and represent it in a mathematical form. The statement "Angles X and Y are supplementary" can be directly translated into the equation X + Y = 180°. This equation forms the foundation of our solution. The second piece of information, "Angle X is 3 times the measure of angle Y," can be expressed as X = 3Y. Now we have two equations with two unknowns, a classic setup for solving a system of equations. The beauty of this problem lies in its ability to test our understanding of both geometric concepts and algebraic techniques. By skillfully manipulating these equations, we can isolate the variables and determine the measures of angles X and Y.

To solve this problem effectively, the crucial step is translating the given information into mathematical equations. This process involves identifying the key relationships and expressing them using variables and symbols. In this case, we have two key pieces of information: the supplementary nature of angles X and Y, and the relationship between their measures.

The first piece of information, "Angles X and Y are supplementary," directly translates into the equation: X + Y = 180°. This equation represents the fundamental definition of supplementary angles, where the sum of their measures equals 180 degrees. It serves as the cornerstone of our solution, providing a direct link between the two unknown angles. The second piece of information, "Angle X is 3 times the measure of angle Y," can be expressed as: X = 3Y. This equation establishes a proportional relationship between the two angles, stating that the measure of angle X is three times that of angle Y. Now, we have a system of two equations with two unknowns, which can be solved using various algebraic techniques. This step of translating words into equations is a fundamental skill in mathematics, allowing us to represent abstract concepts in a concrete and manipulable form. The ability to accurately set up equations is paramount to solving mathematical problems effectively.

Now that we have established our system of equations: X + Y = 180° and X = 3Y, the next step is to solve for the unknowns, X and Y. There are several algebraic methods we can employ, but substitution is a particularly elegant approach in this case. Since we already have an expression for X in terms of Y (X = 3Y), we can substitute this expression into the first equation.

Substituting X = 3Y into X + Y = 180°, we get: 3Y + Y = 180°. Combining like terms, we have: 4Y = 180°. To isolate Y, we divide both sides of the equation by 4: Y = 180° / 4, which simplifies to Y = 45°. Now that we have the measure of angle Y, we can easily find the measure of angle X using the equation X = 3Y. Substituting Y = 45° into this equation, we get: X = 3 * 45°, which simplifies to X = 135°. Therefore, the measure of angle X is 135 degrees and the measure of angle Y is 45 degrees. This systematic approach of substitution allows us to unravel the unknowns step by step, leading us to the solution.

After solving the system of equations, we have determined that the measure of angle X is 135 degrees and the measure of angle Y is 45 degrees. The question specifically asks for the measure of angle X. Therefore, based on our calculations, the measure of angle X is 135°. It's always a good practice to double-check our answer to ensure it aligns with the given information and makes logical sense within the context of the problem.

In this case, we can verify our solution by confirming that angles X and Y are indeed supplementary. Adding their measures, we get 135° + 45° = 180°, which confirms that they are supplementary angles. We can also check if angle X is three times the measure of angle Y: 135° = 3 * 45°, which is also true. These verifications provide confidence in the accuracy of our solution. Therefore, the final answer to the problem, "What is the measure of angle X?" is 135 degrees. This problem serves as a clear illustration of how algebraic techniques can be applied to solve geometric problems effectively.

In conclusion, by carefully dissecting the problem statement, setting up equations, and employing algebraic techniques, we have successfully determined the measure of angle X to be 135 degrees. This problem highlights the importance of understanding the definition of supplementary angles and the ability to translate geometric relationships into algebraic equations. The systematic approach we used, involving substitution and verification, is a valuable problem-solving strategy applicable to a wide range of mathematical problems.

This exercise not only reinforces our understanding of supplementary angles but also hones our problem-solving skills in mathematics. The ability to translate word problems into mathematical equations is a crucial skill, and this problem provides a clear example of how to do so effectively. Moreover, the verification step underscores the importance of checking our answers to ensure accuracy and logical consistency. By mastering concepts like supplementary angles and developing robust problem-solving techniques, we can confidently tackle more complex geometric challenges in the future.

The answer is D. 135°