Construct 4 pairs of parallel lines in different orientations.

Question

Homi · Accepted Answer

Understand the Question

Parallel lines can be constructed using a ruler and compass by making corresponding angles equal along a transversal line.
To construct $4$ pairs of parallel lines in different orientations, the same construction method is applied across horizontal, vertical, and two opposite diagonal directions.

Step 1 · Draw a Line and a Transversal

Draw a straight line $AB$ . Draw another line $CD$ that intersects $AB$ at point $E$ . Line $CD$ serves as the transversal. Diagram 1

Step 2 · Choose a Point on the Transversal

Mark any point $P$ on the transversal line $CD$ (where $P \neq E$ ). We will construct a line passing through $P$ that is parallel to $AB$ . Diagram 2

Step 3 · Copy the Angle at Point E to Point P

With $E$ as the center and any convenient radius, draw an arc cutting line $AB$ at $F$ and line $CD$ at $G$ .
Keeping the same compass radius, with $P$ as the center, draw an arc cutting $CD$ at $H$ .
Set the compass opening equal to the distance $FG$ .
With $H$ as the center and radius equal to $FG$ , draw an arc cutting the previous arc at point $I$ .

Step 4 · Draw the Parallel Line

Join points $P$ and $I$ and extend the line on both sides to get line $JK$ .

Since $\angle PEB = \angle CPK$ (corresponding angles are equal), line $JK \parallel AB$ . Diagram 4

Step 5 · Repeat for Different Orientations

Repeat the same construction steps to draw $4$ pairs of parallel lines in different orientations:

Pair 1 (Horizontal): Base line drawn horizontally, parallel line constructed above or below.
Pair 2 (Vertical): Base line drawn vertically, parallel line constructed beside it.
Pair 3 (Diagonal — Slanting Right): Base line drawn with a positive slope, parallel line constructed along a transversal.
Pair 4 (Diagonal — Slanting Left): Base line drawn with a negative slope, parallel line constructed along a transversal.

Answer

$4$ pairs of parallel lines are constructed in horizontal, vertical, and two diagonal orientations using the corresponding angles method.

Common Mistakes

Changing Compass Radius: Altering the compass width between drawing the arc at $E$ and the arc at $P$ will result in unequal corresponding angles, causing the lines to intersect instead of being parallel.
Incorrect Arc Measurement: Measuring $FG$ incorrectly leads to an inaccurate intersection point $I$ .

Question 15

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