Parallel lines and a transversal

Two parallel lines can be cut by a third line called a transversal. The cutting produces eight angles , and many of them turn out to be equal or supplementary.

Idea

Draw two parallel lines $l \parallel m$ and a third line $t$ cutting both. At each crossing, four angles form. Number them $1, 2, 3, 4$ at the first crossing and $5, 6, 7, 8$ at the second (same positions).

These rules hold only when the two cut lines are parallel. In fact, the converse also holds: if any one of the above relations is observed, the lines are parallel.

So, in practice, if you spot any of these, you can deduce the others. For instance, if $\angle 1 = 70^\circ$ and the two cut lines are parallel, then $\angle 5 = 70^\circ$ (corresponding), $\angle 4 = 110^\circ$ (linear pair with $\angle 1$), $\angle 8 = 70^\circ$ (vertically opposite $\angle 5$), and so on.

Why these relations hold. Geometrically, sliding one parallel onto the other moves the four angles at one crossing exactly onto the four at the other , so corresponding angles must coincide. The other relations follow by combining this with linear-pair and vertical-angle facts.

Worked examples

Example 1. Parallel lines $l \parallel m$ are cut by $t$. One angle is $60^\circ$. Find its corresponding angle.

Corresponding angles are equal at parallels: $60^\circ$.

Example 2. Same setup. The angle is $60^\circ$. Find the co-interior angle on the same side.

Co-interior angles sum to $180^\circ$, so the partner is $180 - 60 = 120^\circ$.

Example 3. If alternate interior angles are $(2x + 10)^\circ$ and $(3x - 30)^\circ$, find $x$ and the angle.

Alternate angles at parallels are equal: $2x + 10 = 3x - 30 \Rightarrow x = 40$. The angle: $2(40) + 10 = 90^\circ$.

Example 4. Two lines are cut by a transversal. The pair of corresponding angles are $55^\circ$ and $55^\circ$. Are the two lines parallel?

Yes , by the converse, equal corresponding angles guarantee parallels.

Try it yourself

1. In the picture, $\angle 1 = 70^\circ$ and $l \parallel m$. Find $\angle 5$ (corresponding).
2. With the same data, find the co-interior partner of $\angle 1$.
3. Alternate interior angles are $(x + 25)^\circ$ and $(3x - 15)^\circ$ with $l \parallel m$. Find $x$.
4. If $\angle 1 = 110^\circ$ and $l \parallel m$, find $\angle 2$ (its linear-pair partner) and $\angle 6$ (corresponding to $\angle 2$).
5. Are lines with $\angle$corresponding $= 80^\circ$ and $\angle$ corresponding $= 100^\circ$ parallel?
6. Co-interior angles on the same side are $x$ and $2x$. Find $x$.
7. Mark all eight angles in a clear figure. If $\angle 1 = 65^\circ$, write down all eight (assuming parallels).
8. State the converse of the corresponding-angle rule.

Activity

Draw two parallel lines on grid paper (use grid rows to ensure parallelism). Pick any transversal and label all eight angles. Measure each with a protractor and verify the four relationships above.