The problem requires us to find the supplementary angle to a given angle of $45^\circ$. A supplementary angle is an angle that, when added to the given angle, sums up to $180^\circ$.

1. Problem Description

The problem requires us to find the supplementary angle to a given angle of $45^\circ$. A supplementary angle is an angle that, when added to the given angle, sums up to $180^\circ$.

2. Solution Steps

Let the given angle be $x = 45^\circ$.

Let the supplementary angle be $y$.

We know that the sum of supplementary angles is $180^\circ$. Therefore, we have:

$x + y = 180^\circ$

Substitute the given value of $x$:

$45^\circ + y = 180^\circ$

Subtract $45^\circ$ from both sides of the equation to solve for $y$:

$y = 180^\circ - 45^\circ$

$y = 135^\circ$

3. Final Answer

The supplementary angle is $135^\circ$.