Math PicnicEvery sheet is a promise

Copy an angle, step by step

An angle is copied without ever knowing how many degrees it is.

Practice Practice sheetNew questions
The question
Copy angle ABC onto the ray from D.ACBD

Copy angle ABC onto the ray from D.

Copy angle ABC onto the ray from D.

compass+straightedge
Step 1→With the point on B, draw an arc crossing both arms of the angle
Step 2→Keeping that opening, draw the same arc from D, crossing the ray
Step 3→Set the compass to the distance between the two crossings on the first arc
Step 4→Step that distance round from where the second arc meets the ray, and draw the arm through it
arcs=the evidence
  1. Copy angle ABC onto the ray from D. Two tools only — a compass that holds a distance and a straightedge that draws a line. Nothing here is measured.
  2. 1. With the point on B, draw an arc crossing both arms of the angle. The arc crosses both arms, which turns the angle into two points a fixed distance from the vertex. An angle is hard to carry; two points are not.
  3. 2. Keeping that opening, draw the same arc from D, crossing the ray. The same opening at D gives the same arc there, so the new figure starts out matching the old one at that radius.
  4. 3. Set the compass to the distance between the two crossings on the first arc. The distance across the first arc between the two crossings is the last thing the angle depends on. Carry it with the compass.
  5. 4. Step that distance round from where the second arc meets the ray, and draw the arm through it. Stepping that distance round the second arc builds a triangle with the same three sides as the first — SSS — so the angle between the arms has to be equal too.
  6. Leave every arc showing. The arcs are the evidence the figure was constructed rather than eyeballed, and a marker reads them to check the method as well as the result.

How it works.

  1. 01

    Copy angle ABC onto the ray from D. Two tools only — a compass that holds a distance and a straightedge that draws a line. Nothing here is measured.

  2. 02

    1. With the point on B, draw an arc crossing both arms of the angle. The arc crosses both arms, which turns the angle into two points a fixed distance from the vertex. An angle is hard to carry; two points are not.

  3. 03

    2. Keeping that opening, draw the same arc from D, crossing the ray. The same opening at D gives the same arc there, so the new figure starts out matching the old one at that radius.

  4. 04

    3. Set the compass to the distance between the two crossings on the first arc. The distance across the first arc between the two crossings is the last thing the angle depends on. Carry it with the compass.

  5. 05

    4. Step that distance round from where the second arc meets the ray, and draw the arm through it. Stepping that distance round the second arc builds a triangle with the same three sides as the first — SSS — so the angle between the arms has to be equal too.

  6. 06

    Leave every arc showing. The arcs are the evidence the figure was constructed rather than eyeballed, and a marker reads them to check the method as well as the result.

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