Solving for the Angle: Supplementary and Complementary Angles Ratio

Understanding the relationship between supplementary and complementary angles is fundamental in geometry. This article explores how to solve for an angle given the ratio of its supplementary and complementary angles. We'll use a step-by-step approach, starting with the basics, and apply algebraic techniques to find the measure of the angle.

Basics of Supplementary and Complementary Angles

A supplementary angle is formed when two angles add up to 180 degrees. If x is the measure of an angle, its supplementary angle is 180 - x degrees.

A complementary angle, on the other hand, is formed when two angles add up to 90 degrees. Similarly, the complementary angle of x is 90 - x degrees.

The Problem Context

The problem at hand is to find the measure of an angle given that the ratio of its supplementary angle to its complementary angle is 13:4. Let's denote the angle as x degrees. We know that:

Supplementary angle: 180 - x Complementary angle: 90 - x

According to the problem, the ratio can be expressed as:

(frac{180 - x}{90 - x} frac{13}{4})

Step-by-Step Solution

Let's solve this step-by-step:

Start with the given ratio:

(frac{180 - x}{90 - x} frac{13}{4})

Cross-multiply to get rid of the fractions:

(4(180 - x) 13(90 - x))

Expand both sides:

(720 - 4x 1170 - 13x)

Rearrange the equation to isolate x:

(720 - 1170 -13x 4x)

(-450 -9x)

(x frac{450}{9})

(x 50)

Conclusion

The measure of the angle is 50 degrees.

Additional Insights

The steps outlined above reflect a systematic approach to solving such problems. Notably, expressing ratios as differences (13:4 -> 130:40) can simplify the problem, but the algebraic approach provides a general solution.

Understanding these concepts is crucial for solving more complex geometric problems and is especially useful in fields such as architecture, engineering, and design.