Transversals: corresponding and alternate angles

What is a transversal?

Sometimes it is hard to tell by eye whether two lines are parallel. Transversals settle it. A line t that crosses two lines l and m is a transversal, and it makes eight angles: ∠1 to ∠4 where it meets l, and ∠5 to ∠8 where it meets m. Can all eight be different? No. At each crossing the vertically opposite angles are equal, so ∠1 = ∠3, ∠2 = ∠4, ∠5 = ∠7 and ∠6 = ∠8: four equal pairs, and at most four different measures.

Corresponding angles

The transversal makes one set of angles with l and another with m. Angles in the same position in the two sets, such as ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, and ∠4 and ∠8, are corresponding angles.

Activity 3 asks for a pair of lines and a transversal that make only two different angles. Draw l and a transversal t meeting at X, and suppose the angle at X is 60°; its linear partner is then 120°. Mark a point Y on t and draw a line m through Y that also makes 60° with t, copying the angle with tracing paper or a protractor. Lines l and m come out parallel. So when the corresponding angles made by a transversal are equal, the lines are parallel.

The other way round is also true. If a transversal crosses two parallel lines, trace one angle on tracing paper and slide it to the corresponding angle: it matches exactly. Corresponding angles made by a transversal across parallel lines are always equal. And if two lines are not parallel, as in Activity 5, no transversal can make a pair of corresponding angles equal.

Alternate angles

Now take parallel lines l and m and a transversal t, with angles a, b, c, d at l and e, f, g, h at m. ∠f and ∠d form a pair of alternate angles, and so do ∠e and ∠c: they lie between the lines on opposite sides of the transversal. If ∠f is 120°, what is ∠d? ∠b corresponds to ∠f, so ∠b = 120°, and ∠d is vertically opposite ∠b, so ∠d = 120°. The same steps work for any measure of ∠f, so alternate angles made by a transversal across parallel lines are always equal.

Interior angles on the same side

In Example 3, parallel lines are crossed by a transversal and ∠3 is 50°. Then ∠2 = 180° − 50° = 130°, a linear pair with ∠3, and ∠6 = ∠2 = 130° as corresponding angles. Angles such as ∠3 and ∠6, which lie between the lines on the same side of the transversal, are interior angles, and 50° + 130° = 180°. This always happens: ∠2 + ∠3 = 180° and ∠2 = ∠6, so ∠3 + ∠6 = 180°.

Using the angle facts

With one angle known, the rest follow. In Example 1, ∠6 is 135°: then ∠2, ∠4 and ∠8 are 135° too, through corresponding and vertically opposite angles, and the other four are 180° − 135° = 45°. In Example 2, ∠a is 120° and ∠f is 70°; then ∠b = 60°, but ∠b and ∠f are corresponding angles and are not equal, so the lines are not parallel.

In Example 4, ABCD has AB parallel to CD and AD parallel to BC, with ∠DAC = 65° and ∠ADC = 60°. With AD as the transversal across AB and CD, the interior angles give ∠DAB = 180° − 60° = 120°, so ∠CAB = 120° − 65° = 55°. With CD as the transversal across AD and BC, ∠BCD = 180° − 60° = 120°, and in the same way ∠ABC = 60°.

Why it matters

Transversals give a reliable test for parallel lines, and they let one angle reveal many others. The angles at a single crossing are explained in linear pairs and vertically opposite angles, and drawing parallel lines in parallel and perpendicular lines. All six Figure it Out questions are solved in the NCERT solutions for Chapter 5; see all of the Chapter 5 notes.