3.3 Proving lines parallel or perpendicular

Proving Lines Parallel or Perpendicular

When you encounter two lines cut by a transversal, the angle relationships tell you whether those lines are parallel. And when two lines meet at exactly 90°, you can prove they're perpendicular. These proofs show up constantly in geometry, so getting comfortable with the underlying theorems is worth the effort.

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Proving Lines Parallel

To prove two lines are parallel, you need to show that a specific angle relationship exists when a transversal crosses both lines. There are three main approaches, each based on a different angle pair.

Methods of Proving Parallel Lines

Corresponding Angles Converse: Corresponding angles sit in the same relative position at each intersection ($\angle 1$ and $\angle 5$, $\angle 2$ and $\angle 6$). If corresponding angles are congruent, then the lines are parallel.

Alternate Interior Angles Converse: Alternate interior angles are on opposite sides of the transversal, between the two lines ($\angle 3$ and $\angle 6$, $\angle 4$ and $\angle 5$). If alternate interior angles are congruent, then the lines are parallel.

Same-Side Interior Angles Converse (Co-Interior): Same-side interior angles are on the same side of the transversal, between the two lines ($\angle 3$ and $\angle 5$, $\angle 4$ and $\angle 6$). If same-side interior angles are supplementary (sum to $180°$), then the lines are parallel.

Notice the pattern: for corresponding and alternate interior angles, you need congruence. For same-side interior angles, you need supplementary. Mixing these up is one of the most common mistakes on proofs.

Proving Lines Perpendicular

Key Properties of Perpendicular Lines

Two lines are perpendicular if and only if they intersect to form right angles ($90°$). You only need to show that one of the four angles at the intersection is $90°$, since the others follow automatically by the Linear Pair Postulate and Vertical Angles Theorem.

A useful related theorem: if a transversal is perpendicular to one of two parallel lines, then it's perpendicular to the other as well. This connects the parallel and perpendicular concepts and shows up frequently in proofs.

Applying the Theorems in Proofs

When writing a proof involving parallel or perpendicular lines, follow this general approach:

1. Identify the given information. What angle relationships or line properties are you told?
2. Determine what you need to prove. Are you showing lines parallel, perpendicular, or both?
3. Select the right theorem. Match the given angle pair type to the correct converse theorem.
4. Write the logical chain. State each step with its justification (given, theorem name, or definition).

For example, if you're given that $\angle 1 \cong \angle 5$ and these are corresponding angles formed by a transversal cutting lines $l$ and $m$, you can conclude $l \parallel m$ by the Converse of the Corresponding Angles Postulate.

Construction of Parallel and Perpendicular Lines

Constructing a parallel line through a point not on a given line:

1. Draw a transversal from the given point to the line, creating an angle of intersection.
2. At the given point, copy that angle on the same side of the transversal (constructing a congruent corresponding angle using compass and straightedge).
3. The ray through the copied angle forms a line parallel to the original, justified by the Converse of the Corresponding Angles Postulate.

Constructing a perpendicular line depends on where the point is:

Point on the line:

1. From the point, swing equal arcs on both sides along the line to mark two points equidistant from your original point.
2. From each of those two points, swing arcs of equal radius above (or below) the line so they intersect.
3. Draw a line from that intersection through the original point. This line is perpendicular to the given line.

Point off the line:

1. From the point, swing an arc that crosses the line at two places.
2. From each of those two intersection points, swing equal arcs on the opposite side of the line so they intersect.
3. Draw a line from the original point through that new intersection. This line is perpendicular to the given line.

In both cases, the construction works because you're creating points equidistant from a center, which forces a $90°$ angle by symmetry.