Parallel Lines and a Transversal

This is the workhorse lesson of the chapter. Imagine two parallel lines and a third line , a transversal , slicing across them. The transversal makes eight angles with the two lines. When the two lines are parallel, these eight angles fall into a tight pattern: most are equal, and the rest sum to $180^\circ$. Knowing this pattern lets you find any angle once one is given.

Definitions

Two lines in a plane are parallel if they never intersect, no matter how far they are extended. We write $\ell \parallel m$.

A transversal is a third line that crosses both $\ell$ and $m$ at two distinct points.

When a transversal cuts $\ell$ at $P$ and $m$ at $Q$, the eight angles formed are grouped:

• Corresponding angles: same-side, same-position pairs at $P$ and $Q$. There are four such pairs.
• Alternate interior angles: between the two lines, on opposite sides of the transversal. Two pairs.
• Alternate exterior angles: outside the two lines, on opposite sides of the transversal. Two pairs.
• Co-interior (same-side interior) angles: between the two lines, on the same side of the transversal. Two pairs.

The four key theorems

Theorem 1 (Corresponding angles). If $\ell \parallel m$ and a transversal cuts them, then each pair of corresponding angles is equal.

Theorem 2 (Alternate interior angles). If $\ell \parallel m$ and a transversal cuts them, then each pair of alternate-interior angles is equal.

Theorem 3 (Co-interior angles). If $\ell \parallel m$ and a transversal cuts them, then each pair of co-interior angles sums to $180^\circ$.

Theorem 4 (Alternate exterior angles). If $\ell \parallel m$ and a transversal cuts them, then each pair of alternate-exterior angles is equal.

These four theorems are not independent; you can deduce any one from any other (using the linear pair axiom and the vertically-opposite-angles theorem). For exam purposes, treat them as four named facts.

Proof idea for Theorem 1 (the "corresponding" version). This is taken as one of the consequences of the parallel postulate (or as an axiom, depending on the textbook). Once Theorem 1 is established, the others follow:

• Alternate interior $=$ corresponding via vertically opposite angles at one intersection.
• Co-interior sum $=$ linear pair at one intersection $+$ corresponding equality yields $180^\circ$.
• Alternate exterior $=$ corresponding via vertically opposite angles at the other intersection.

The eight-angle pattern

Label the eight angles $1, 2, 3, 4$ at $P$ (upper line) and $5, 6, 7, 8$ at $Q$ (lower line), going round each intersection in the same direction. A common labelling:

• At $P$: $\angle 1$ upper-left, $\angle 2$ upper-right, $\angle 3$ lower-right, $\angle 4$ lower-left.
• At $Q$: $\angle 5$ upper-left, $\angle 6$ upper-right, $\angle 7$ lower-right, $\angle 8$ lower-left.

Then:

• Corresponding: $(\angle 1, \angle 5), (\angle 2, \angle 6), (\angle 3, \angle 7), (\angle 4, \angle 8)$.
• Alternate interior: $(\angle 3, \angle 5), (\angle 4, \angle 6)$.
• Co-interior: $(\angle 3, \angle 6), (\angle 4, \angle 5)$.
• Alternate exterior: $(\angle 1, \angle 7), (\angle 2, \angle 8)$.

If you are told any one of the eight, the other seven are determined. Four of them equal the given one; four equal $180^\circ$ minus the given one.

The converse

Each theorem above has a converse, equally useful.

Converse of Theorem 1. If a transversal cuts two lines such that a pair of corresponding angles is equal, then the two lines are parallel.

Converse of Theorem 2. If a transversal cuts two lines such that a pair of alternate-interior angles is equal, then the two lines are parallel.

Converse of Theorem 3. If a transversal cuts two lines such that a pair of co-interior angles sums to $180^\circ$, then the two lines are parallel.

These let us prove parallelism: show that one of the angle relationships holds, and conclude $\ell \parallel m$.

Worked examples

Example 1. Two parallel lines are cut by a transversal making an angle of $50^\circ$ at the upper line. Find all eight angles.

The corresponding angle at the lower line is also $50^\circ$. Linear-pair partners are $130^\circ$. Vertically opposite are $50^\circ$. By symmetry, four angles are $50^\circ$ and four are $130^\circ$.

Example 2. Two lines are cut by a transversal making alternate-interior angles of $70^\circ$ and $70^\circ$. Are the lines parallel?

Yes , by the converse of the alternate-interior-angles theorem.

Example 3. In a figure, $\ell \parallel m$ and the transversal makes a co-interior pair of $4x^\circ$ and $(x + 40)^\circ$. Find $x$.

Co-interior angles sum to $180^\circ$: $4x + x + 40 = 180 \Rightarrow 5x = 140 \Rightarrow x = 28$.

Example 4. $\ell \parallel m$. A transversal makes an angle of $108^\circ$ with $\ell$. Find the corresponding angle on $m$, and the co-interior partner.

Corresponding angle: $108^\circ$. Co-interior partner: $180 - 108 = 72^\circ$.

Example 5. Lines $\ell$ and $m$ are crossed by a transversal. The corresponding angles are $(2x + 10)^\circ$ and $(3x - 30)^\circ$. For what value of $x$ are $\ell$ and $m$ parallel?

By the converse of the corresponding-angles theorem, $\ell \parallel m$ iff $2x + 10 = 3x - 30 \Rightarrow x = 40$.

Try it yourself

1. State the corresponding-angles theorem.
2. State the alternate-interior-angles theorem.
3. State the co-interior-angles theorem.
4. State the converse of each of the three above.
5. Two parallel lines are cut by a transversal. One angle is $60^\circ$. Find the other seven.
6. A transversal cuts two lines making alternate-interior angles of $50^\circ$ and $50^\circ$. Are the lines parallel?
7. $\ell \parallel m$ and the transversal makes a co-interior pair $(3x - 5)^\circ$ and $(2x + 10)^\circ$. Find $x$.
8. If alternate-exterior angles equal $40^\circ$ and $40^\circ$, are the lines parallel?
9. A transversal makes a corresponding pair of $2x^\circ$ and $80^\circ$ with two parallel lines. Find $x$.
10. Sketch two parallel lines cut by a transversal and label all eight angles.

Pitfalls / Insight

• Identify the type before applying. Mistaking alternate-interior for co-interior swaps equality with sum-to-$180^\circ$.
• The theorems require the lines to be parallel. Without parallelism, the relations need not hold.
• The converse goes from angle equality to parallelism. Always identify which direction you need.

Insight. A transversal of two parallel lines determines an eight-angle pattern of just two values: $a$ and $180^\circ - a$. Once you can recognise corresponding, alternate, and co-interior pairs at sight, every problem in the chapter reduces to spotting the pattern and reading off the value.