In these lessons, we will learn how to construct a 45° angle by bisecting a 90-degree angle.

**Related Pages**

Constructing Angles

Construct 30° Angles

More Geometry Lessons

We can use the angle bisector method to construct some other angles from existing angles.

**Examples:**

A 30° angle can be obtained by bisecting a 60° angle.

A 15° angle can be obtained by bisecting a 30° angle.

A 45° angle can be obtained by bisecting a 90° angle.

A 22.5° angle can be obtained by bisecting a 45° angle.

**How to construct a 45° angle by bisecting a 90° angle?**

**Example:**

The figure shows a point *A* on a straight line. Construct an angle of 45° at point *A*.

**Solution:**

Construct a 90° angle, and then construct an angle bisector to obtain a 45° angle.

Step 1: Stretch your compasses until it is more then half the length of
*AB*. Put the sharp end at *A* and mark an arc above and another arc below line segment *AB*.

Step 2: Without changing the width of the compasses, put the sharp end
at *B* and mark arcs above and below the line segment *AB* that will intersect with the arcs drawn in step 1.

Step 3: Join the two points where the arcs intersect with a straight line.
This line is the perpendicular bisector of *AB*. P is the midpoint of *AB*.

Step 4: Bisect the 90-degree angle to form a 45-degree angle.

In summary the steps to construct a 45° angle are:

- Construct perpendicular lines to obtain a 90° angle.
- Bisect the 90° angle to obtain a 45° angle.

**How to construct a 45-degree angle using compass and straightedge?**

Step 1: Construct perpendicular lines (90 degrees).

Step 2: Bisect that angle to get a 45-degree angle.

**How to construct a 30-degree angle without using a protractor?**

This video shows two ways to construct a 30-degree angle.

The first way starts by constructing part of an equilateral triangle, then bisecting the 60° angle.

The second method starts by constructing a rhombus with 60° and 120° angles, then joining the
opposite vertices to leave the 30° angle.

**How to construct a 75-degree angle with a compass?**

Step 1: Construct 60-degree angles by constructing an equilateral triangle.

Step 2: Bisect a 60-degree angle to form a 30-degree angle.

Step 3: Bisect the 30-degree angle to form a 15-degree angle.

Step 4: Combine a 60-degree angle with a 15-degree angle to form a 75-degree angle.

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