91Ó°ÊÓ

Understanding angles is essential in geometry. An angle is formed by two rays with a common endpoint, and it measures the rotation needed to align one ray with the other. Angles are typically measured in degrees, with a full circle containing 360 degrees (\(360^\text{°}\)). Complementary angles are two angles whose sum is 90 degrees (\(90^\text{°}\)). This means if one angle is known, its complement can be found by subtracting the angle from 90 degrees. For example, if one angle measures \(30^\text{°}\), its complement is \(90^\text{°} - 30^\text{°} = 60^\text{°}\). Understanding complementary angles is crucial for solving various geometric problems.

Linear equations are a type of algebraic equation where the highest power of the variable is one. They are used to represent relationships between quantities and can always be written in the form \(ax + b = c\). In the context of the given problem, we set up a linear equation to find the measure of an angle based on its complement. The problem statement gives us: 'the complement of the angle is five times its measure.' Using this information, we can formulate the equation: \(90^\text{°} - x = 5x\). Here, \(x\) represents the unknown angle, and the equation allows us to solve for \(x\) systematically. Linear equations are foundational in mathematics as they appear in various applications ranging from simple to complex scenarios.

Verifying solutions is a critical step to ensure the accuracy of the result. After solving an equation, it's important to substitute the solution back into the original problem to check if it satisfies all given conditions. In our problem, we solved for \(x\) and found it to be \(15^\text{°}\). To verify, we calculate the complement of \(15^\text{°}\), which is: \(90^\text{°} - 15^\text{°} = 75^\text{°}\). Then, we check if this complement is indeed five times the measure of our angle: \(5 \times 15^\text{°} = 75^\text{°}\). Since both calculations match, our solution is verified. This process not only confirms the correctness but also reinforces understanding of the problem.