Bisecting Segments and Angles

Lesson 2 of 4: Geometric Construction

In this lesson:

• Copy a segment exactly — no ruler reading
• Bisect a segment and an angle with a compass
• Construct 60°, 90°, 120°, 30°, and 45°

What Angles Does a Builder Need?

Look at a set square. What angle is this corner? What about this one?

• The large corner: 90°
• The small corners: 60° and 30° (on one type)
• Or: 45° and 45° (on the other type)

These are the exact angles a draughtsman uses for a building plan.

Can a Protractor Guarantee Exactness?

A draughtsman needs an exact 60° corner on a plan.

• Marks 60° with his protractor: reads to the nearest degree
• Shifts it slightly and re-marks: reading changes by 1°
• A slipped protractor gives a slipped angle

A pair of compasses guarantees exactness.

The Compass Captures a Length — Exactly

Copy cm from .

Ruler: Shift it: reading changes.

Compass:

1. Point on , pencil on — captures length
2. Same opening; point on ; arc →
3. — no ruler reading

The compass opening is the length.

The Arc Primitive Drives Every Construction

Every construction today is one idea:

Fix a radius → strike arcs from two centres → use where they cross

Shared radius → crossings are exactly symmetric → construction is exact.

All five constructions in this lesson are this primitive, applied.

Copying a Segment: Full Procedure

Copy cm from .

1. Point on , pencil on — capture the radius
2. Base line; mark
3. Arc from cutting the line at
4. — by construction

The compass carried the length. No ruler needed.

Your Turn: Copy a Segment

Draw segment (5–8 cm) in your exercise book.

1. Set your compass to
2. Draw a base line; mark point
3. Arc from → mark
4. Measure — does it equal ?

Aim for a difference less than 0.5 mm.

The Perpendicular Bisector: Two Jobs in One

• Job 1: Cuts at its midpoint — equal halves
• Job 2: Meets at exactly 90°

Building the Perpendicular Bisector: Step by Step

Perpendicular bisector of cm.

1. Open compass to more than half of — roughly 5 cm
2. Point on : arc above and arc below
3. Same opening; point on : arcs crossing the first two
4. Join the two crossings → this is the bisector

Why Equal Radii Force Symmetry

Both crossings are equidistant from and .

• Any point equidistant from two endpoints lies on the perpendicular bisector
• Both crossings satisfy this — above and below
• Joining them traces the unique bisector-and-perpendicular

Equal radii force the crossings to sit symmetrically — that is exactness.

Your Turn: Construct a Perpendicular Bisector

Draw cm. The arc from is done — open to 4.5 cm, above and below.

You complete:

• Same opening from — cross the arcs above and below
• Join the two crossing points
• Check: both halves = 3.5 cm; angle = 90°

Verify: Both Jobs Done Correctly

After constructing the perpendicular bisector of cm:

• Measure the left half: 3.5 cm ✓
• Measure the right half: 3.5 cm ✓
• Measure the angle at the crossing: 90° ✓

One construction. Both results. Protractor used for checking, not for building.

Spot the Error: Which Construction Is Correct?

Two students attempted the perpendicular bisector.

• Construction A: Equal radii from and — crossings above and below
• Construction B: Compass re-opened for arcs — crossing shifts right

Which gives the true midpoint and 90°? Why?

Commit to your answer before continuing.

Same Word, Different Starting Centre

Which centre you start from changes everything.

Same primitive. Different starting centres.

Bisecting an Angle: The Three Steps

1. Given : point on vertex : arc cutting both arms — crossings on , on
2. Same opening from : arc in interior; same from : arcs cross at
3. Draw ray — this is the angle bisector

Bisecting 80°: The Construction Worked

Bisect .

• Draw angle (protractor)
• Arc from (4 cm): on , on
• Same radius from , → cross at
• Ray : each half = 40°

Check:

Your Turn: Bisect a 110° Angle

Draw . The arc from is given — on , on .

1. Equal arcs from and into the interior — mark crossing
2. Draw ray

Check: each half = 55°. What is your reading?

Check Both Bisection Angles Now

Verify your work with a protractor:

• (half of 80°) = 40° ✓
• (other half) = 40° ✓
• Each half of 110° = 55° ✓

Protractor checks the construction. Arcs did the building.

60°: Built from an Equilateral Arc

• Arc from (radius ) marks
• Same radius from : arc crossing at
• Three equal sides → equilateral → each angle 60°

The arcs build the triangle without drawing it.

Standard Angles: The Full Table

All five: arc primitive, no protractor.

Construct a 45° Angle — On Your Own

Task: Construct a 45° angle from scratch on a fresh line.

• No step-by-step scaffold
• Choose your construction path (which two constructions do you need?)
• Use only compass and straight edge
• Check the result with a protractor

Plan your steps first, then execute.

Three Errors That Break Constructions

Error 1 — Changing the opening mid-construction
→ Set the radius; lock it; don't touch the screw until all arcs are done.

Error 2 — Drawing with a protractor instead of constructing
→ Protractor checks; compass builds.

Error 3 — Starting from the wrong centre
→ Segment: endpoints. Angle: vertex.

One Primitive, Four Constructions Today

• Copy any segment — no ruler reading
• Midpoint AND perpendicular — one construction
• Bisect any angle — one construction
• Build 60°, 90°, 120°, 30°, 45°
• Next: 60° → equilateral triangle; 90° → square

Fix radius → arcs from two centres → use the crossing