Understanding Supplementary Angles

In geometry, two angles are considered supplementary if their measures add up to exactly 180 degrees. Think of them as forming a straight line when placed adjacent to each other.

If we have an angle, let's call it $\theta$, its supplementary angle is the angle that, when added to $\theta$, results in 180 degrees. So, the supplementary angle to $\theta$ is $180^\circ - \theta$.

Setting Up the Angle Problem

The question tells us that an angle is $\frac{3}{7}$ times its supplementary angle. Let the unknown angle be $\theta$ (measured in degrees). The supplementary angle to $\theta$ is $(180 - \theta)$ degrees.

According to the problem statement, we can write the following equation:

The angle ($\theta$) is equal to $\frac{3}{7}$ multiplied by its supplementary angle ($(180 - \theta)$).

Mathematically, this is expressed as:

\(\theta = \frac{3}{7} \times (180 - \theta)\)

Solving the Equation for the Angle

Now we need to solve this equation to find the value of $\theta$.

1. Start with the equation: \(\theta = \frac{3}{7}(180 - \theta)\)
2. Multiply both sides of the equation by 7 to get rid of the fraction: \(7 \times \theta = 7 \times \frac{3}{7}(180 - \theta)\)
3. This simplifies to: \(7\theta = 3(180 - \theta)\)
4. Distribute the 3 on the right side: \(7\theta = 3 \times 180 - 3 \times \theta\)
5. This gives: \(7\theta = 540 - 3\theta\)
6. Now, we want to gather all terms involving $\theta$ on one side. Add \(3\theta\) to both sides of the equation: \(7\theta + 3\theta = 540 - 3\theta + 3\theta\)
7. This simplifies to: \(10\theta = 540\)
8. Finally, isolate $\theta$ by dividing both sides by 10: \(\theta = \frac{540}{10}\)
9. Calculate the result: \(\theta = 54\)

So, the angle is 54 degrees.

Verifying the Supplementary Angle Calculation

Let's check our answer. If the angle is $54^\circ$, its supplementary angle is $180^\circ - 54^\circ = 126^\circ$.

The problem states the angle is $\frac{3}{7}$ times its supplementary angle. Let's calculate $\frac{3}{7}$ of $126^\circ$:

\(\frac{3}{7} \times 126^\circ = 3 \times \frac{126}{7}^\circ = 3 \times 18^\circ = 54^\circ\)

Our calculated angle, $54^\circ$, matches the value we obtained from the equation. This confirms our solution is correct.

Therefore, the angle is 54°.

Revision Table: Types of Angles

Additional Information on Angles and Geometry

Understanding the relationships between angles is fundamental in geometry. Besides supplementary angles, another important concept is complementary angles, which add up to 90 degrees. Many geometric problems involve setting up algebraic equations based on these angle relationships, similar to how we solved this problem involving supplementary angles and a given ratio.

When tackling angle problems:

• Identify the type of angle or angle relationship involved (e.g., supplementary, complementary, vertically opposite, angles on a straight line, angles around a point).
• Assign variables to unknown angles.
• Use the definitions of angle relationships to write equations.
• Solve the algebraic equations to find the unknown angle values.
• Always double-check your answer by plugging the value back into the original problem statement or diagram.

Practice with different types of angle problems will help you become more comfortable with setting up and solving these geometric equations.