Geometry A hub · 01.02

Basic Constructions

The compass copies a length. Keep the width, or the copy is fake.

Congruent means the measures match

≅ is the sign. Matching hash marks on two segments mean they are congruent. Different hash counts mean they are not. Do not trust “they look the same.”

A construction copies a figure with only a compass and a straightedge. No ruler numbers. No protractor. Euclid’s rule, still the rule on the quiz.

The compass opening is a radius. When you lock that width and swing, every point on the arc is the same distance from the pointy end. That is why a copied segment is congruent: you copied a radius.

Paint is not a construction program. Dynamic geometry (circles that stay circles when you drag a point) counts. A freehand drawing app does not.

Copy a segment

You are given AB. You want a matching length somewhere else.

  1. Draw a ray with one endpoint, longer than AB. Call the endpoint C.
  2. Open the compass to AB.
  3. Park it on C. Swing an arc that cuts the ray. That cut is D.
  4. CD ≅ AB.
AB CD

To copy AB you have to identify the distance between A and B. Not a midpoint. Not a random extra point.

The usual miss: the arc from C is a different width than AB. Then it is not a copy, even if a ray was drawn neatly.

Copy an angle

Same idea: copy a distance, twice. First you copy how far the arc sits from the vertex. Then you copy the chord between the two rays.

  1. Given ∠ABC. Draw a new ray with endpoint D. D is the new vertex.
  2. Compass on B. Arc through both rays. Hits at E and F.
  3. Same width, compass on D. Matching arc. Hits the new ray at G.
  4. Open the compass to EF (the chord).
  5. Park it on G. Arc that cuts the new arc at H.
  6. Ray DH. Now ∠HDG ≅ ∠ABC.
B C A D

Spot this on a quiz: you see the original angle with an arc through both rays, and a lone ray somewhere else. That is copying an angle, mid-job. If the second arc is coming from a point on the first arc, not from a new vertex, you have switched to an angle bisector.

Bisect a segment

Bi- = two. Sect = cut. Equal pieces. The cut hits the midpoint. The line you draw is also perpendicular to the segment. Same construction, two names: segment bisector and perpendicular bisector.

  1. Compass on one endpoint. Open more than halfway.
  2. Swing an arc above and below the segment.
  3. Keep the width. Other endpoint. Arcs above and below that cut the first two.
  4. Line through the two intersection points. That line hits the midpoint.
ABM
Width trap: after the first pair of arcs, do not pinch or spread the compass before you move to the other endpoint. That is the error the quiz writes into a student-step list.

Next step after “compass on A, open more than half, swing above and below”: same width, compass on B, swing above and below so the arcs meet.

Bisect an angle

Same cut-in-half idea. The two smaller angles are congruent.

  1. Compass on the vertex. Arc that cuts both rays. Mark those two hits.
  2. Compass on the first hit. Arc inside the angle.
  3. Same width, compass on the second hit. Arc that cuts the first inner arc.
  4. Ray from the vertex through that inner intersection.
A

Wrong picture of an angle bisector: the two inner arcs were swung from the vertex again, or the compass width changed between the two inner arcs, so they do not meet in a fair spot.

What is this drawing doing?

This is the headline skill for 01.02. The quiz shows a mid-construction sketch and asks you to name it. Tap the name. Four drawings rotate.

Practice like 01.02

Ten questions. Diagram ID, next-step, “who changed the compass,” order of copy-an-angle. New items.