Copying a segment and copying an angle are two basic compass-and-straightedge constructions in geometry. They matter because they create congruent figures without measuring with a ruler or protractor. A copied segment has the same length as the original, and a copied angle has the same angle measure as the original.
These constructions build the foundation for triangles, perpendicular bisectors, angle bisectors, and many formal proofs.
To copy a segment, set the compass width to the original segment and transfer that width from a new starting point on a ray or line. To copy an angle, draw a new ray, copy an arc from the original angle, then copy the chord distance between the arc intersections to locate the second side of the new angle. The straightedge is used only to draw lines and rays, while the compass transfers distances exactly.
Congruence is verified by matching compass widths, arc marks, and corresponding endpoints or rays.
Key Facts
- Congruent segments have equal lengths: AB = CD.
- Congruent angles have equal measures: m∠ABC = m∠DEF.
- A compass transfers distance without using a number scale.
- A straightedge draws straight lines, rays, and segments but does not measure length.
- To copy segment AB from point C, set the compass to AB and mark point D so that CD = AB.
- To copy ∠ABC, copy one arc from the vertex, then copy the distance between the two arc intersection points.
Vocabulary
- Congruent
- Two geometric figures are congruent if they have the same size and shape.
- Segment
- A segment is a part of a line with two endpoints.
- Angle
- An angle is formed by two rays that share a common endpoint called the vertex.
- Compass
- A compass is a tool used to draw arcs or circles and to transfer distances.
- Straightedge
- A straightedge is a tool used to draw straight lines but not to measure distance.
Common Mistakes to Avoid
- Changing the compass width while copying a segment is wrong because the transferred length will no longer match the original segment.
- Using ruler numbers to copy the segment is wrong because a compass-and-straightedge construction should transfer distance directly, not by measurement.
- Placing the copied angle arc at the wrong vertex is wrong because the arc must be centered on the new angle vertex to preserve the angle opening.
- Copying only the arc radius for an angle is wrong because the distance between the two arc intersection points must also be copied to locate the second ray.
Practice Questions
- 1 Segment AB is 7.5 cm long. You copy it from point C using a compass and mark point D. What should the length of CD be?
- 2 An angle measures 42 degrees. You construct a congruent copy of it using only a compass and straightedge. What is the measure of the copied angle?
- 3 Explain why copying the distance between the two arc intersection points is necessary when constructing a congruent copy of an angle.