Lesson 13.1.1

Copying Segments & Angles

Using only a compass and straightedge, you can create an exact copy of any segment or angle — no ruler measurements needed.

Introduction

Classical constructions use only two tools: a compass (for drawing arcs) and a straightedge (for drawing lines). No measuring — only transferring distances and intersecting arcs.

Past Knowledge

Segment and angle properties. Congruence (Unit 5). Circle properties.

Today's Goal

Copy a segment and copy an angle with compass and straightedge.

Future Success

Bisectors (13.1.2), parallel lines (13.1.3), inscribed polygons (13.1.4–13.1.5).

Key Concepts

The Two Tools

  • Compass: Draws circles/arcs. Transfers distances without measuring.
  • Straightedge: Draws straight lines through two points. Has no markings.

Step-by-Step Constructions

Copying a Segment

  1. 1Draw a ray from a new endpoint.
  2. 2Set compass width to the original segment length.
  3. 3Place compass on the new endpoint and draw an arc crossing the ray.
  4. 4The intersection point creates a congruent segment.

Copying an Angle

  1. 1Draw a ray for the new angle's side.
  2. 2From the original angle's vertex, draw an arc crossing both sides.
  3. 3Draw the same arc from the new vertex.
  4. 4Measure the chord distance between the original arc intersections.
  5. 5Transfer that chord to the new arc. Draw the ray through the intersection.

Worked Examples

Why It Works (Angle Copy)

The compass sets equal radii, creating two congruent triangles (SSS) between the original and new arcs. Congruent triangles → congruent angles.

∎ SSS congruence of the arc triangles guarantees an exact copy.

Common Pitfalls

Changing Compass Width Mid-Step

Don't adjust the compass between steps unless instructed. The whole point is to transfer a fixed distance.

Real-Life Applications

Architecture & Drafting

Before CAD software, architects used compasses and straightedges to replicate precise angles and distances on blueprints.

Stained Glass & Mosaic Art

Artists use compass constructions to create precise geometric patterns for stained glass windows.

Practice Quiz

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