Formal Constructions: Perpendicular and Parallel Lines

Lesson 2 of 2: Perpendicular, Parallel, Alternative Tools

In this lesson:

• Construct perpendicular and parallel lines
• Justify constructions using congruence and postulates
• Perform constructions with paper folding, Mira, and GeoGebra

Learning Objectives for Lesson Two

By the end of this lesson:

1. Perform all six constructions — perpendicular and parallel included
2. Explain why each construction works using definitions and theorems
3. Use alternative tools: paper folding, reflective devices, GeoGebra
4. Justify each result with precise geometric definitions

Lesson One Built Your Construction Toolkit

From Lesson 1:

• Copy segment (circle definition)
• Copy angle (SSS: )
• Bisect segment (equidistance); bisect angle (SSS)

Today's new constructions use these:

• Perpendicular on a line → segment bisection
• Parallel through external point → angle copying

Perpendicular Through a Point on a Line

1. Arc at → marks , on with
2. Equal arcs at and → intersect at
3. Draw — perpendicular to at

Why Perpendicular Through P Works

This reduces to bisecting segment from Lesson 1:

• Step 1: → is midpoint of
• Steps 2–3: perpendicular bisector of
• Perpendicular bisector passes through midpoint at

Line at ✓

Check: Which Lesson One Construction Appears Here?

Steps 2 and 3 of the perpendicular-through-P construction reproduce which Lesson 1 procedure?

Name it before advancing.

• (a) Copy a segment
• (b) Copy an angle
• (c) Bisect a segment — perpendicular bisector
• (d) Bisect an angle

Answer: This Is Segment Bisection

Answer: (c) — bisect a segment (perpendicular bisector)

• Step 1 sets as midpoint of
• Steps 2–3 = exact perpendicular bisector procedure from Lesson 1
• Bisector passes through at right angle to

Each new construction assembles earlier ones in a clever order.

Dropping a Perpendicular from Outside

1. Large arc at → crosses at and
2. Construct perpendicular bisector of
3. Bisector passes through ; meets at

Why Dropping the Perpendicular Works

and on arc centered at →

Any point equidistant from and lies on the perpendicular bisector of

lies on the bisector → bisector passes through and meets at ✓

Constructing a Parallel Line Through a Point

1. Transversal through crossing at
2. Copy angle at to point
3. New ray through is the parallel line

Why the Parallel Construction Works

Angle-copying creates congruent corresponding angles:

• Transversal meets at , forming angle
• Angle-copy at →
• Congruent corresponding angles → lines parallel

Converse Corresponding Angles Postulate → Line ✓

Watch Out: Eyeballing Parallelism Fails

Mistake: drawing a line that looks parallel, skipping angle-copy.

• Visually parallel lines often diverge when extended
• "Eyeballing" cannot guarantee congruent corresponding angles
• The angle-copy step is the geometric guarantee

The construction exploits the definition of parallel lines directly.

Check: Which Step Forces the Lines Parallel?

Which step in the parallel construction activates the Corresponding Angles Converse?

Think before advancing.

• (a) Drawing the transversal through
• (b) Noting the angle at
• (c) Copying the angle at to match
• (d) Extending the ray at into a full line

Answer: Copying the Angle Creates Parallelism

Answer: (c) — copying the angle at

• Steps 1–2: setup only — no parallelism yet
• Step 3: creates equal corresponding angles — the guarantee
• Step 4: extends the result — not the cause

Paper Folding Creates the Same Bisectors

• Fold onto → crease is perpendicular bisector
• Fold one ray onto the other → crease bisects angle

Why: Folding is a reflection. Fold-line points are equidistant from and — that is the perpendicular bisector.

Reflective Devices Visualize the Symmetry

A Mira is a transparent mirror placed on the paper:

Perpendicular bisector:

• Adjust Mira until reflection of lands on
• Mira's edge = perpendicular bisector

Angle bisector:

• Adjust Mira until one ray reflects onto the other
• Mira's edge = angle bisector

GeoGebra Replicates Compass and Straightedge

Three digital tools mirror the two physical tools:

• Circle with center through point → compass
• Line through two points → straightedge

Unique advantage: Drag any point — construction updates dynamically, proving constructions are general.

All Methods Exploit the Same Geometry

Different tools — same geometric properties.

Key Takeaways: All Six Constructions

• Copy segment → circle definition; copy angle → SSS
• Bisect segment → equidistance; bisect angle → SSS
• Perp on line → bisect ; perp from point → equidistance
• Parallel → angle copy + Converse Corresponding Angles

Compass, folding, Mira, GeoGebra — same geometry.

Coming Up: Inscribing Regular Polygons

HSG.CO.D.13 applies all six constructions:

• Equilateral triangle inscribed in a circle
• Square inscribed in a circle — uses perpendicular bisector
• Regular hexagon — side equals the circle's radius

All six constructions from these lessons are the tools.