In geometry, understanding the properties and relationships between angles is crucial. One such relationship is that of complementary and supplementary angles. Complementary angles are two angles whose measures add up to 90 degrees, while supplementary angles add up to 180 degrees. When given the ratio of the angles, we can solve for the exact measures using algebraic methods. In this article, we explore a specific problem where the ratio of two complementary angles is given as 2/7, and we determine the measure of the supplementary angle of the smaller angle.

Introduction to Complementary and Supplementary Angles

Complementary angles are a pair of angles whose measures add up to 90 degrees. For example, if one angle is 30 degrees, its complementary angle is 60 degrees. On the other hand, supplementary angles are a pair of angles whose measures add up to 180 degrees. If one angle is 50 degrees, its supplementary angle is 130 degrees.

Given Problem: Ratio of Complementary Angles is 2/7

The problem at hand is to find the measure of the supplementary angle of the smaller angle, given that the ratio of the two complementary angles is 2/7. Let's denote the two complementary angles as x and y. According to the problem, the ratio of these angles is:

$$frac{x}{y} frac{2}{7}$$

Since x and y are complementary angles, we also know:

$$x y 90^circ$$

Solving for the Angles

To find the values of x and y, we start by expressing x in terms of y using the given ratio:

$$x frac{2}{7}y$$

We substitute x in the complementary angle equation:

$$frac{2}{7}y y 90^circ$$

Next, we combine the terms by converting y to a fraction:

$$frac{2}{7}y frac{7}{7}y 90^circ$$

This simplifies to:

$$frac{2 7}{7}y 90^circ$$

$$frac{9}{7}y 90^circ$$

Now, we solve for y by multiplying both sides by the reciprocal of (frac{9}{7}):

$$y 90^circ times frac{7}{9} 70^circ$$

With y found, we determine x by substituting y back into the equation:

$$x 90^circ - y 90^circ - 70^circ 20^circ$$

Therefore, the smaller angle is x 20°.

Finding the Supplementary Angle of the Smaller Angle

To find the supplementary angle of the smaller angle, we use the fact that the sum of supplementary angles is 180 degrees:

$$180^circ - x 180^circ - 20^circ 160^circ$$

Hence, the measure of the supplementary angle of the smaller angle is:

$$boxed{160^circ}$$

Conclusion

This problem demonstrates the application of basic algebra and the properties of complementary and supplementary angles. Understanding these principles is foundational in geometry and can be applied in various real-world scenarios, such as engineering, architecture, and design.