Question

An angle is greater than $45°$. Is its complementary angle greater than $45°$ or equal to $45°$ or less than $45°$?

Answer 1

Complementary Angles.

When the sum of the measures of two angles is $90°$, then the angles are called complementary angles. In other words, if two angles add up to form a right angle, then these angles are referred to as complementary angles.

Thus, two angles $\alpha$ and $\beta$ are said to be complementary angles if $\alpha +\beta =90°$.

Let us assume the complementary angles be $x$ and $y$ where it is given that $x>45°$.

Since the sum of the measures of complementary angles pair is $90°$, therefore

$x+y=90°$

We have,

$x>45°$

Add $y$ on both the sides of the above inequality:

$\begin{array}{rcl}x+y& >& 45°+y\\ & \Rightarrow & 90°>45°+y\left[\because x+y=90°\right]\\ & \Rightarrow & 90°-45°>y\\ & \Rightarrow & 45°>y\\ & \Rightarrow & y<45°\end{array}$

Therefore, the complementary angle of $x$ is less than $45°$.

Hence, the complementary angle of an angle which is greater that $45°$ is always less than $45°$.