Understanding the concepts of complementary and supplementary angles is crucial in geometry. Let’s break down these concepts and see how we can find the supplement of a complement.
Complementary Angles
Complementary angles are two angles whose measures add up to 90 degrees. For instance, if one angle measures 30 degrees, its complement is 60 degrees, because $30^text{°} + 60^text{°} = 90^text{°}$
Supplementary Angles
Supplementary angles are two angles whose measures add up to 180 degrees. For example, if one angle measures 110 degrees, its supplement is 70 degrees, because $110^text{°} + 70^text{°} = 180^text{°}$
Finding the Supplement of a Complement
Now that we understand complementary and supplementary angles, let’s find the supplement of a complement.
Step-by-Step Process
- Identify the Angle: Suppose we have an angle $x$
- Find the Complement: The complement of angle $x$ is $90^text{°} – x$
- Find the Supplement of the Complement: The supplement of the complement is $180^text{°} – (90^text{°} – x)$
Simplifying the Expression
Let’s simplify the expression $180^text{°} – (90^text{°} – x)$:
$180^text{°} – 90^text{°} + x = 90^text{°} + x$
So, the supplement of the complement of angle $x$ is $90^text{°} + x$
Example
Let’s take an example to make this clearer.
Suppose we have an angle of 40 degrees.
- Find the Complement: The complement of 40 degrees is $90^text{°} – 40^text{°} = 50^text{°}$
- Find the Supplement of the Complement: The supplement of 50 degrees is $180^text{°} – 50^text{°} = 130^text{°}$
Using our formula $90^text{°} + x$, we get $90^text{°} + 40^text{°} = 130^text{°}$, which matches our previous calculation.
Conclusion
To find the supplement of a complement, remember the formula $90^text{°} + x$. This formula simplifies the process and ensures you get the correct result every time. Understanding these relationships between angles can help you solve more complex geometric problems with ease.
3. CK-12 Foundation – Complementary and Supplementary Angles