To construct a $135^\circ$ angle, you can follow a systematic approach that begins with creating a $90^\circ$ angle and then bisecting it to form a $45^\circ$ angle.

Step 1: Draw the Baseline

Start by drawing a straight line using a ruler; this will serve as your baseline.

Step 2: Create a $90^\circ$ Angle

Next, use a protractor to measure and mark a $90^\circ$ angle from the baseline. From this vertex, draw a line that extends through the mark you made, thereby forming a right angle.

Step 3: Bisect the $90^\circ$ Angle

To create a $45^\circ$ angle, position the protractor on the vertex of the $90^\circ$ angle. Measure and mark $45^\circ$ on one of the arms of the right angle. Draw a line from the vertex through this $45^\circ$ mark; this line bisects the $90^\circ$ angle, resulting in two $45^\circ$ angles.

Step 4: Construct the $135^\circ$ Angle

To achieve a $135^\circ$ angle, you need to combine the $90^\circ$ angle with one of the $45^\circ$ angles. Place the protractor on the vertex of the original $90^\circ$ angle and measure $135^\circ$ from the baseline. Mark this point, and then draw a line from the vertex through the mark you just made. This new line will form a $135^\circ$ angle with the baseline.

By following these steps, you can accurately construct a $135^\circ$ angle using basic geometric tools such as a ruler and a protractor. This method not only ensures precision but also enhances your understanding of the relationships between different angles.