Given below are the steps to construct a 135 degree angle:

First mark a Point 'O' and then draw a straight line with 'o' as the starting point.

Name that line as OA. Now, taking point O as the center, draw an arc with a compass.

Place the compass at two end points of the arc and observe that meeting point of two arcs represents the measure of 90 degrees.

Draw a dotted line, which will represent the perpendicular line on the given line OA. This dotted line divides the straight angle into two halves. We can also say that this line is the angle bisector of straight angle.

Put the compass at arc touching 90 degrees and 180 degrees and mark the arc at the middle of two angles. Meeting point of two arcs represents the angle of measure 135 degrees.

Join the point 'O' with point of intersection of two arcs with a sharpened pencil. Name this point as B.

The angle AOB is having the measure of 135 degrees.
We know that half of 90 degree is 45 degree. So, we say that 90 + 45 = 135 degrees.
135 degrees is represented as angle bisector of 90 degrees and 180 degrees.