To construct a 45° angle by first constructing a perpendicular bisector of a segment and then bisecting one of the angles, follow these steps:

Steps:

1. Draw a Segment: Draw a line segment $AB$ of any length using a ruler.

2. Construct the Perpendicular Bisector:

   • Place the compass point on point $A$ and draw an arc above and below the line segment $AB$ with a radius greater than half the length of the segment.
   • Without changing the radius, place the compass point on point $B$ and draw another arc above and below $AB$, intersecting the previous arcs.
   • Label the points of intersection of the arcs above and below $AB$ as $P$ and $Q$.
   • Draw a straight line through points $P$ and $Q$. This is the perpendicular bisector of $AB$, which forms a 90° angle with $AB$ at its midpoint $M$.

3. Label the Intersection Point: The perpendicular bisector intersects segment $AB$ at its midpoint $M$, creating two equal halves of the segment and forming a 90° angle.

4. Bisect the 90° Angle:

   • To bisect the 90° angle formed between the perpendicular bisector and segment $AB$, place the compass point at point $M$ and draw an arc that intersects both the perpendicular bisector and $AB$.
   • Label the points of intersection as $C$ (on the perpendicular bisector) and $D$ (on segment $AB$).
   • Now, place the compass point on $C$ and draw an arc inside the 90° angle. Without changing the compass width, place the compass point on $D$ and draw another arc inside the 90° angle, intersecting the previous arc. Label this intersection point as $E$.
   • Draw a straight line from point $M$ through point $E$. This line bisects the 90° angle, forming two 45° angles.

Now, the angle formed between $AB$ and the bisector line $ME$ is 45°.

Summary:

• Step 1: Draw a segment.
• Step 2: Construct its perpendicular bisector.
• Step 3: Bisect the 90° angle to get a 45° angle.

Would you like more details on any of these steps?

---

Related Questions:

1. How can you verify that the constructed angle is exactly 45°?
2. How do you construct a 60° angle using a compass?
3. How can you construct a perpendicular line to a given line without using a protractor?
4. What is the geometrical significance of bisecting angles in triangle constructions?
5. How would you construct a 30° angle by bisecting a 60° angle?

Tip: Always make sure your compass is firm and doesn’t shift when drawing arcs for accurate constructions.