Formal Constructions: Copying and Bisecting

Lesson 1 of 2: Introduction, Copying, Bisecting

In this lesson:

• Distinguish formal construction from drawing
• Copy a segment and copy an angle
• Bisect a segment and bisect an angle

Learning Objectives for Lesson One

By the end of this lesson:

1. Perform: copy a segment, copy an angle, bisect a segment, bisect an angle
2. Explain why each construction works — circle definitions and SSS
3. Distinguish formal construction from a measurement-based drawing
4. Justify each result with geometric definitions

Did Your Ruler Give an Exact Copy?

Try this: copy a 7.3 cm segment using only a ruler.

• Problem: every ruler has manufacturing tolerance
• Problem: parallax error from reading the scale
• Problem: your pencil tip has finite width

A drawing is only as accurate as its tools. A construction is exact by definition.

Only Two Tools Are Permitted Here

Every step is justified by one of Euclid's two postulates.

Construction Versus Drawing: Key Differences

Constructions Are Proofs in Action

Every construction step is a logical statement:

• "Place compass at A through B" → creates the circle with center A, radius AB
• "All points on this arc" → are exactly distance AB from A
• "Mark point Q on the arc" → (by circle definition)

The Compass Preserves Distance Exactly

The compass does one thing perfectly:

• Set to any width between two points
• Every point it marks is exactly that distance from the center
• Transfer width elsewhere → exact copy, no measurement needed

Think of the compass as "copy-paste for distances."

How to Copy a Segment Step by Step

Step 1: Ray from in any direction
Step 2: Set compass: point at , pencil at
Step 3: Pivot at , sweep arc → mark

Circle Definition Guarantees the Copy

Circle centered at , radius : every point at distance from .

• Compass width = ; arc at has radius
• on this arc → ✓

No measurement — the definition does the work.

How to Copy an Angle Step by Step

1. Ray from ; arc at → and on both rays
2. Same arc at → marks
3. Set compass to ; arc at → marks
4. Draw ray

SSS Congruence Justifies Angle Copying

• (same arc radius)
• (same compass setting, step 3)
• (same arc radius)

SSS: → ✓

Check: Which Step Forces Angle Congruence?

Which step in the angle-copy construction is the critical one for SSS?

Think about the three side pairs before advancing.

• (A) Drawing the initial ray from
• (B) Drawing the first arc centered at
• (C) Setting the compass to distance
• (D) Drawing ray

Answer: Setting Compass to GH Is Critical

Answer: (C) — set compass to and mark

• Steps 2+3 give and (two sides)
• Step C gives — the third side for SSS

Each construction step establishes one geometric fact.

Watch Out: Copying Is Not Measuring

Mistake: verifying a copy by measuring with a ruler.

• Construction → exact result by definition
• Ruler → approximate reading of that exact result
• Disagreement? The ruler is wrong, not the construction.

How to Bisect a Segment Step by Step

1. Set compass ; arc at
2. Same width; arc at
3. Label intersections (above), (below)
4. Draw line — perpendicular bisector meeting at

Equidistance Proves the Bisector Works

on both arcs → ; same logic:

Any point equidistant from and lies on the perpendicular bisector of .

and both qualify → line is the perpendicular bisector ✓

Watch Out: Any Width Greater Than Half Works

Mistake: setting compass to exactly .

• Any width works — arcs will intersect
• The midpoint is determined by equidistance, not compass radius
• Arcs not intersecting? Open wider.

How to Bisect an Angle Step by Step

1. Arc at → marks on ray , on ray
2. Arc at with radius ; same arc at
3. Arcs intersect at ; draw ray

SSS Congruence Justifies Angle Bisection

(SSS):

→ bisects ✓

Check: What Rule Must Step Three Follow?

After the arc at , what must be true of step 3's radius?

Write your answer before advancing.

• (a) Same radius as step 1
• (b) Compass to distance
• (c) Same radius as step 2
• (d) Compass to distance

Answer: Step Three Must Match Step Two

Answer: (c) — step 3 radius must equal step 2

• Step 2: arc at , radius →
• Step 3: arc at , same → → ✓
• Different radii → triangles not congruent → bisection fails

Paper Folding Achieves the Same Result

Bisecting a segment by folding:

• Fold so lands exactly on
• Crease = perpendicular bisector of

Why: Folding is a reflection mapping . Every fold-line point is equidistant from and — exactly the perpendicular bisector.

Same geometry, different tool.

Key Takeaways: Copying and Bisecting

✓ Construction = exact; drawing = limited by tool precision
✓ Copy segment: circle definition →
✓ Copy angle: SSS →
✓ Bisect segment: equidistance; bisect angle: SSS

Bisection width: any works
Arc intersections are off the segment

Coming Up in Lesson Two

Lesson 2 covers the remaining three constructions:

• Perpendicular through a point on a line
• Perpendicular from an external point to a line
• Parallel line through an external point

Plus: paper folding, Mira, and GeoGebra alternatives.