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Angle45Detailed.de0114128377dc7f5fb64faea975f4dd.png

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:

  1. Construct line AD

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°.

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