Instruction: Construct an angle of 45° with the given side.

Goal: 2L 5E

Available tools:

• Move
• Point
• Line
• Circle
• Perpendicular Bisector
• Perpendicular
• Angle Bisector
• Intersect

Pack: Gamma

Previous level: 3.6

Next level: 3.8

## 2L solution

Let A be the initial point of the given ray and B a distinct arbitrary point on that ray.

1. Construct the perpendicular to the ray from A; let C be a distinct arbitrary point on that perpendicular
2. Construct the angle bisector of ∠BAC

## 5E solution

Let A be the initial point of the given ray and B a distinct arbitrary point on that ray.

1. Construct the perpendicular to the ray from B
2. Construct the circle with center B and radius AB, intersecting the perpendicular at C and D
3. Construct line AC

Alternate 5E solution:

1. Construct the circle with center A and radius AB
2. Construct the circle with center B and radius AB, intersecting circle A at C and D
3. Construct line BC, intersecting circle B at E
4. Construct the circle with center E and radius AE, intersecting line BC at F
5. Construct line AF

## 2V solution

From the 2L solution:

1. Construct the angle bisector of ∠BAD, where D is an arbitrary point on line AC on the side opposite of C from A

From the 5E solution:

From the alternate 5E solution:

1. Construct a circle with an arbitrary center G on the ray and radius FG
2. Construct a circle with another arbitrary center H on the ray and radius GH, intersecting the previous circle at I
3. Construct line AI

## Explanation

By tracing the perpendicular to the line, the angle bisector will divide the 90° angle into two 45° angles.

The 5E solution has you construct a right isosceles triangle, which has two angles of 45°.