Problem 63 - Algebra 2

Question

Given: Two angles are complementary The measure of one angle is \(15^{\circ}\) more than the measure of the other angle. Conclusion: The measures of the angles are \(30^{\circ}\) and \(45^{\circ} .\) This conclusion - F is contradicted by the first statement given. G is verified by the first statement given. H invalidates itself because a \(45^{\circ}\) angle cannot be complementary to another. J verifies itself because \(30^{\circ}\) is \(15^{\circ}\) less than \(45^{\circ}\) .

Step-by-Step Solution

Answer

F is contradicted by the first statement given.

Step 1: Understand the Definition of Complementary Angles

Two angles are said to be complementary if the sum of their measures is equal to \(90^{\circ}\). This means when you add the measures of the two angles, you should get \(90^{\circ}\).

Step 2: Define Variables for the Angles

Let's define the measure of one angle as \(x\). Then, based on the problem statement, the measure of the other angle would be \(x + 15^{\circ}\), since it is \(15^{\circ}\) more than the first angle.

Step 3: Set up the Equation for Complementary Angles

As the angles are complementary, their sum should equal \(90^{\circ}\). Thus, we can write the equation: \[ x + (x + 15) = 90 \]

Step 4: Solve the Equation

Simplify the equation from Step 3:\[ x + x + 15 = 90 \] Combine like terms: \[ 2x + 15 = 90 \]Subtract \(15\) from both sides to solve for \(x\): \[ 2x = 75 \]Divide by \(2\):\[ x = 37.5 \]

Step 5: Find the Measure of the Second Angle

Now that we know \(x = 37.5\), substitute back to find the measure of the second angle:\[ x + 15 = 37.5 + 15 = 52.5 \]

Step 6: Conclusion Verification

The solutions give angle measures of \(37.5^{\circ}\) and \(52.5^{\circ}\), not \(30^{\circ}\) and \(45^{\circ}\). Therefore, the conclusion given in the problem is not correct based on our calculations.

Key Concepts

Angle Measurement

Understanding angle measurement is the key to solving many geometry problems. Angles are typically measured in degrees, denoted by the symbol \(^{\circ}\). The size of an angle can range from \(0^{\circ}\) to \(360^{\circ}\), where \(0^{\circ}\) represents no angle and \(360^{\circ}\) is a complete circle.

In the context of complementary angles, we need to focus on angles that add up to \(90^{\circ}\). These are two angles whose measures, when summed, give a right angle. The classical example is when one angle is \(30^{\circ}\) and the other is \(60^{\circ}\), since together they total \(90^{\circ}\).

When working with problems that involve measuring angles, it's crucial to pay attention to the given values and the relationships described among them. In this case, one angle is described as being \(15^{\circ}\) more than another, which directly leads to setting up an equation to find their actual measures.

Algebraic Equations

Algebra plays a vital role in solving geometry problems like finding angle measures. An algebraic equation uses letters to represent unknown values, allowing us to set up relationships and find solutions.

For example, if one angle in a complementary pair is known to be \(15^{\circ}\) more than another, you express this as an equation by defining one angle as \(x\) and the other as \(x + 15^{\circ}\). Using the property of complementary angles, you then know that \(x + (x + 15) = 90\).

This equation can be solved step by step:

• Simplify to combine like terms: \(2x + 15 = 90\).
• Subtract \(15\) from both sides: \(2x = 75\).
• Divide by \(2\): \(x = 37.5\), determining the first angle.

The measure of the second angle is then found by plugging back \(x = 37.5\) into \(x + 15 = 52.5\). This process shows how algebra helps in figuring out the measures of both angles based on their relationship.

Problem-Solving Process

Approaching a mathematical problem systematically is essential for accuracy and understanding. Here, solving for complementary angles involves recognizing key information, translating it into mathematical expressions, and methodically executing calculations.

The process begins by understanding what complementary angles are, verified by their well-known sum of \(90^{\circ}\). When given additional details, such as one angle being \(15^{\circ}\) more than the other, the next step is setting up an algebraic equation reflecting that relationship.

Solving the equation involves orderly steps, ensuring each operation is performed with care: simplifying the equation, handling expressions properly, and performing arithmetic correctly. This disciplined approach ensures results are accurate and logical, revealing the measures of \(37.5^{\circ}\) and \(52.5^{\circ}\), not the initially assumed \(30^{\circ}\) and \(45^{\circ}\).

Each step in the problem-solving process builds upon the last, guiding us to a result that communicates whether the initial conditions stated were valid or not. This method is what brings clarity and resolution to mathematical challenges.