Problem 52 - Algebra 2

Question

REVIEW Given: Two angles are supplementary. One angle is \(25^{\circ}\) more than the measure of the other angle. Conclusion: The measures of the angles are \(65^{\circ}\) and \(90^{\circ} .\) This conclusion \(-\) \(\mathrm{F}\) is contradicted by the first statement given. \(\mathrm{G}\) is verified by the first statement given. H invalidates itself because a \(90^{\circ}\) angle cannot be supplementary to another. J verifies itself because \(90^{\circ}\) is \(25^{\circ}\) more than \(65^{\circ} .\)

Step-by-Step Solution

Answer

The conclusion is contradicted (\(H\)) because given angles don't sum to \(180^{\circ}\).

Step 1: Understanding Supplementary Angles

Supplementary angles are two angles whose measures add up to \(180^{\circ}\). This means if two angles are supplementary, the sum of their angles must be \(180^{\circ}\).

Step 2: Express the Relationship

Let the measure of one angle be \(x\). According to the problem, the other angle is \(25^{\circ}\) more than this angle. Thus, the other angle can be expressed as \(x + 25^{\circ}\).

Step 3: Set Up the Equation

Since the two angles are supplementary, we can write the equation: \[x + (x + 25^{\circ}) = 180^{\circ}\].

Step 4: Solve for the First Angle \(x\)

Simplify and solve the equation: \[2x + 25^{\circ} = 180^{\circ}\] Subtract \(25^{\circ}\) from both sides: \[2x = 155^{\circ}\] Divide by 2: \[x = 77.5^{\circ}\]

Step 5: Calculate the Second Angle

Now, find the other angle: \[x + 25^{\circ} = 77.5^{\circ} + 25^{\circ} = 102.5^{\circ}\].

Step 6: Evaluate the Given Conclusion

The calculated angles \(77.5^{\circ}\) and \(102.5^{\circ}\) do not match the conclusion (\(65^{\circ}\) and \(90^{\circ}\)). The conclusion provided is not verified by the given statements.

Step 7: Determine Which Statement Applies

The statement "H invalidates itself because a \(90^{\circ}\) angle cannot be supplementary to another" is correct; a \(90^{\circ}\) angle and another \(90^{\circ}\) angle would sum to \(180^{\circ}\), but \(65^{\circ} + 90^{\circ}\) is not \(180^{\circ}\). Therefore, the angles suggested must invalidate this fact.

Key Concepts

Understanding Angle Measures

Angles are fundamental building blocks in geometry. They measure how much one ray has rotated around a fixed point, usually referred to as the vertex. The measure of an angle is usually expressed in degrees. A complete rotation is divided into 360 degrees.

When dealing with problems involving angles, it's important to identify the relationship between them. Two angles are described as supplementary if the sum of their measures is exactly 180 degrees. This relationship is crucial for solving exercises like the one given in the problem. For example, if one angle is obtained by rotating a ray slightly from a standard position, and another angle continues from this to complete a semi-circle, together, they form supplementary angles. This is the foundational understanding needed to tackle the equations associated.

• Key Fact: Supplementary angles sum up to 180 degrees.
• Example: If one angle is 120 degrees, the other must be 60 degrees to be supplementary.

Solving the Equation for Angles

Equation solving in geometry involves setting up relationships between unknown values and known constants. In this exercise, we have two angles which together are supplementary. One angle is said to be 25 degrees more than the other.

To find these angles, we express them in terms of a variable, usually represented by \(x\). Let's say the first angle is \(x\) degrees, then the supplementary angle is expressed as \(x + 25^{\circ}\). With supplementary angles, we form an equation: \(x + (x + 25^{\circ}) = 180^{\circ}\). This equation simplifies to \(2x + 25^{\circ} = 180^{\circ}\). Solving this gives \(x = 77.5^{\circ}\), and adding 25 degrees results in the second angle measuring \(102.5^{\circ}\).

Here’s how you solve it step-by-step:

• Set up the equation using the supplementary angle condition.
• Simplify and solve for \(x\).
• Calculate the second angle by adding 25 degrees to the first angle.

It's important to ensure that solutions adhere to the initial conditions, such as angles being supplementary.

Curriculum of Geometry Concepts

Geometry is filled with various concepts that help us understand the space around us. As we delve into topics like angle relationships, it’s essential to grasp the rules that govern them. For this problem, the idea of supplementary angles is at the core.

Supplementary angles illustrate the larger concept of complementary relationships in geometry. Just like supplementary angles add up to a straight line, complementary angles add up to 90 degrees. Understanding these relationships helps in visualizing and solving more complex geometric problems. They form the baseline for working on geometric proofs and understanding the structure of geometric shapes.

Always remember:

• Angles add depth to geometric figures and impact calculations of perimeters and areas.
• Knowing the difference between supplementary and complementary provides foundational knowledge for tackling angles in any geometric context.