Constructing parallel lines

Knowing what parallel lines are is fun; drawing them precisely is even better. Two methods are common: the set-square slide, and the compass-and-ruler method using equal corresponding angles.

Idea

Set-square method. Take a ruler and a set-square (the triangular tool with one right angle).

1. Place one edge of the set-square along the given line $l$.
2. Place the ruler against another edge of the set-square.
3. Holding the ruler firmly, slide the set-square along the ruler to the desired position.
4. Draw the new line along the edge of the set-square that was against $l$.

Because the set-square has not been rotated , only translated , the new edge is parallel to the original.

Compass method (using equal corresponding angles).

1. From the given point $P$, draw any line $t$ to a point $Q$ on $l$. This is the transversal.
2. With $Q$ as centre, draw an arc cutting $l$ and $t$. Let it hit $l$ at $A$ and $t$ at $B$.
3. With $P$ as centre and the same radius, draw an arc on $t$ (above $P$). Mark where it hits $t$.
4. Measure arc length $AB$ with the compass. From the arc on $t$ (above $P$), mark the same length.
5. Join $P$ with this new mark , that is the required parallel.

The trick: you have copied the angle at $Q$ exactly to $P$. Equal corresponding angles means parallel lines.

Both constructions assume careful, accurate handling. A small wobble means a not-quite-parallel result.

Worked examples

Example 1. Use a set-square to draw a line through point $P$ parallel to a given line $l$.

Place one edge of the set-square along $l$. Bring a ruler against the other edge. Slide the set-square until the first edge passes through $P$. Draw along that edge. Done.

Example 2. Draw two lines $5$ cm apart that are parallel.

Draw line $l$. Place the set-square's edge on $l$. Slide a ruler against the perpendicular edge by $5$ cm. Draw the new line. The two lines are exactly $5$ cm apart everywhere , parallel.

Example 3. Draw a transversal cutting two parallel lines $l$ and $m$, and verify that corresponding angles are equal using a protractor.

Draw a transversal $t$. Measure the corresponding angle at each crossing; they should be equal (within drawing error).

Example 4. Why does the compass method work?

Because we copy the angle at $Q$ on line $t$ to the same direction at $P$. Equal corresponding angles ensure the new line is parallel to $l$.

Try it yourself

1. Draw a line and mark a point $P$ above it. Use a set-square to construct the parallel through $P$.
2. Draw two parallel lines, $4$ cm apart.
3. Using compass, draw a line through a given point parallel to a given line.
4. Why must the slide of the set-square be without rotation to give a parallel?
5. State, in words, the key property used in the compass construction.
6. Draw two lines that you think are parallel; verify using a transversal and angle measurement.
7. Draw a square. Are opposite sides parallel? Why?
8. Find five real-world parallels and one real-world perpendicular pair near you.

Activity

On a sheet of paper, draw a line $l$ and mark a point $P$ above it. Use both methods (set-square and compass) to draw the parallel through $P$. Compare the two constructions , they should give the same line.