How do you construct a 135-degree angle?

Construct a 135-degree angle by first constructing a 90-degree angle. Construct an angle bisector for one side of the right angle. The angle addition theorem allows addition of the resulting 90-degree and 45-degree angles to create the required angle. The construction takes a few minutes using ruler and compass.

1. Draw a line segment

   Use the ruler to draw a line-segment. Label the line-segment with X and Y at its endpoints

2. Construct the 90-degree angle

   Set the compass to a convenient length. Place the point of the compass on point x and draw an arc that cuts through line-segment XY at a new point, P. Keep the point of the compass on point X and draw an arc above the line. Move the point of the compass to point P and draw an arc intersecting the first arc. Label the intersecting point as Z. Connect X to Z and extend the line to create a right angle.

3. Extend line-segment XY to T

   Use the ruler to extend the line segment the opposite direction from Y. Label the new endpoint.

4. Construct the angle-bisector

   Place the point of the compass on point X and draw arcs to intersect line-segments XY and XT. Label these intersections as A and B. Place the point of the compass at point A. Draw a new arc near the center of angle TXZ. Repeat with point B. Label this intersection point F. Draw a line segment from X to F and extend the line. Angle FXY is the required 135-degree angle.