Constructing Parallel Lines

Construction of a Line Parallel to a Given Line through a Point not on it (using corresponding angles or alternate interior angles)

Parallel lines are lines in a plane that are always the same distance apart and never intersect. Constructing a line parallel to a given line through a point not on it is a common geometric task.

The construction relies on the properties of parallel lines intersected by a transversal (a line that intersects two or more other lines). Specifically, if a transversal intersects two lines such that the corresponding angles or the alternate interior angles are equal, then the two lines are parallel.

Method 1: Using Equal Corresponding Angles

This method involves copying an angle formed by the given line and a transversal to create an equal corresponding angle at the given point. Equal corresponding angles guarantee parallel lines.

Given: A line $l$ and a point $P$ not lying on line $l$.

Tools: Compass, Straightedge.

Goal: Construct a line $m$ passing through point $P$ such that $m \parallel l$.

Steps:

1. Draw a Transversal: Draw a line (or ray) $t$ that passes through the given point $P$ and intersects the given line $l$ at any convenient point, say $Q$. This line $t$ acts as a transversal cutting lines $l$ and the line we want to construct through $P$. Identify an angle formed by $l$ and $t$, for example, the angle $\angle PQA$ where $A$ is a point on line $l$ to the right of $Q$. This is a corresponding angle location for $P$.
2. Draw Arc at Intersection Q: With $Q$ as the center and using any convenient compass radius, draw an arc that intersects line $l$ at point $A$ and the transversal $t$ at point $B$.
3. Draw Corresponding Arc at P: Without changing the compass radius used in Step 2, place the pointed end of the compass on point $P$ and draw a similar arc that intersects the transversal $t$. Let this intersection point on $t$ be $C$. This arc should extend towards the side where the parallel line is expected to lie (on the same side of $t$ as point $A$).
4. Measure the Angle Opening: Place the pointed end of the compass on point $A$ (where the arc from $Q$ intersects line $l$). Adjust the compass width so that the pencil tip is exactly on point $B$ (where the same arc intersects the transversal $t$). This compass width represents the 'opening' or the chord length of the angle $\angle AQB$.
5. Transfer the Angle Opening: Without changing the compass width from Step 4, place the pointed end of the compass on point $C$ (where the arc from $P$ intersects the transversal $t$). Draw an arc that intersects the arc drawn from $P$ in Step 3. Label this new intersection point $D$.
6. Draw the Parallel Line: Use the straightedge to draw a straight line passing through the given point $P$ and the newly found point $D$.

The line passing through $P$ and $D$ is the line $m$. This line $m$ is parallel to the given line $l$.

Justification (Method 1 - Corresponding Angles)

Given: Line $l$, point $P$ not on $l$. Line $t$ intersects $l$ at $Q$ and passes through $P$. Line $m$ passes through $P$ and $D$, constructed as described above.

To Prove: $m \parallel l$ (Line $PD \parallel$ Line $AQ$).

Proof:

Consider the construction steps. The arc from $Q$ with radius $r_1$ passes through $A$ on $l$ and $B$ on $t$. The arc from $P$ with the same radius $r_1$ passes through $C$ on $t$. The compass was then set to the distance $AB$. An arc from $C$ with this distance $AB$ intersects the arc from $P$ at $D$.

In effect, we constructed point $D$ such that the distance from $C$ to $D$ is equal to the distance from $A$ to $B$. Both $A$ and $D$ lie on arcs drawn with radius $r_1$ from $Q$ and $P$ respectively, and the chords subtended by the angles at $Q$ ($\angle AQB$) and at $P$ ($\angle CPD$) are equal ($AB = CD$).

This method of transferring angle opening ensures that the angle formed at $P$ by the line $m$ and the transversal $t$, which is $\angle CPD$, is equal to the angle $\angle AQB$ formed by line $l$ and the transversal $t$. That is, $\angle CPD = \angle AQB$.

The angles $\angle CPD$ and $\angle AQB$ are corresponding angles formed by the transversal $t$ intersecting lines $l$ and $m$. Since the corresponding angles are equal ($\angle CPD = \angle AQB$), according to the properties of parallel lines, the lines $l$ and $m$ must be parallel.

Since $\angle CPD$ and $\angle AQB$ are corresponding angles, Line $m \parallel$ Line $l$. Q.E.D.

Method 2: Using Equal Alternate Interior Angles

This method also uses a transversal but copies an angle to create an equal alternate interior angle at point $P$. Equal alternate interior angles guarantee parallel lines.

Given: A line $l$ and a point $P$ not lying on line $l$.

Tools: Compass, Straightedge.

Goal: Construct a line $m$ passing through point $P$ such that $m \parallel l$.

Steps:

1. Draw a Transversal: Draw a line (or ray) $t$ that passes through the given point $P$ and intersects the given line $l$ at any convenient point, say $Q$. Identify an alternate interior angle location at $P$ relative to an interior angle at $Q$. For instance, if we choose the interior angle $\angle PQR$ on line $l$ (where $R$ is a point on $l$ to the left of $Q$), the alternate interior angle location at $P$ will be on the opposite side of $t$ and between line $t$ and the parallel line to be constructed.
2. Draw Arc at Intersection Q: With $Q$ as the center and using any convenient compass radius, draw an arc that intersects line $l$ at point $R$ (forming $\angle PQR$) and the transversal $t$ at point $S$.
3. Draw Alternate Interior Arc at P: Without changing the compass radius used in Step 2, place the pointed end of the compass on point $P$ and draw a similar arc that intersects the transversal $t$ at point $T$. This arc should be drawn on the opposite side of $t$ from where the angle $\angle PQR$ was located.
4. Measure the Angle Opening: Place the pointed end of the compass on point $S$ (where the arc from $Q$ intersects the transversal $t$) and adjust the compass width so that the pencil tip is exactly on point $R$ (where the same arc intersects line $l$). This width represents the opening of $\angle SQR$ (which is part of $\angle PQR$).
5. Transfer the Angle Opening: Without changing the compass width from Step 4, place the pointed end of the compass on point $T$ (where the arc from $P$ intersects the transversal $t$). Draw an arc that intersects the arc drawn from $P$ in Step 3. Label this new intersection point $U$.
6. Draw the Parallel Line: Use the straightedge to draw a straight line passing through the given point $P$ and the newly found point $U$.

The line passing through $P$ and $U$ is the line $m$. This line $m$ is parallel to the given line $l$.

Justification (Method 2 - Alternate Interior Angles)

Given: Line $l$, point $P$ not on $l$. Line $t$ intersects $l$ at $Q$ and passes through $P$. Line $m$ passes through $P$ and $U$, constructed as described above.

To Prove: $m \parallel l$ (Line $PU \parallel$ Line $QR$).

Proof:

Similar to Method 1, the construction ensures that the angle formed at $P$ by the line $m$ and the transversal $t$, which is $\angle TPU$, is equal to the angle $\angle SQR$ formed by line $l$ and the transversal $t$. That is, $\angle TPU = \angle SQR$. Note that $\angle SQR$ is the same as $\angle PQR$ if $S$ lies on $t$ and $R$ lies on $l$ as labelled, and $Q$ is the vertex. The construction essentially copies the angle $\angle PQR$ to the position $\angle TPU$.

The angles $\angle TPU$ and $\angle PQR$ are alternate interior angles formed by the transversal $t$ intersecting lines $l$ and $m$. Since the alternate interior angles are equal ($\angle TPU = \angle PQR$), according to the properties of parallel lines, the lines $l$ and $m$ must be parallel.

Since $\angle TPU$ and $\angle PQR$ are alternate interior angles, Line $m \parallel$ Line $l$. Q.E.D.

Example

Answer: