Lines & Angles • Topic 1 of 3

Parallel Lines & Transversal

When a transversal cuts two parallel lines it creates eight angles that fall into just a few families. Corresponding angles (same position at each intersection, an "F" shape) are equal. Alternate angles — both alternate interior (a "Z" shape) and alternate exterior — are equal. Co-interior angles, also called allied or same-side interior angles (a "C" or "U" shape), are supplementary, summing to 180°. The CAT-smart way to use these: do not memorise eight angle names: instead spot that, because the lines are parallel, every angle in the figure is either equal to a chosen angle x or equal to its supplement 180° − x. Mark one angle x, then label the whole diagram with x and 180° − x and read off the answer. The converse is just as useful in proofs: if a pair of alternate angles is equal, or a pair of co-interior angles sums to 180°, the two lines must be parallel.

✅ Solved examples

1. Two parallel lines are cut by a transversal. One of the angles formed is 65°. Find the co-interior angle paired with it.
Co-interior angles are supplementary: 180° − 65° = 115°.
2. A transversal makes corresponding angles (3x + 10)° and (5x − 30)° on two parallel lines. Find x.
Corresponding angles are equal: 3x + 10 = 5x − 30 ⇒ 40 = 2x ⇒ x = 20.
3. Two parallel lines are cut by a transversal so that a pair of co-interior angles are in the ratio 2 : 3. Find the larger angle.
They sum to 180°: 2k + 3k = 180 ⇒ k = 36 ⇒ larger = 3k = 108°.
4. Lines AB and CD are parallel. A point P lies between them with PE perpendicular to AB. A line from P to AB makes 35° with AB; what angle does it make with CD on the same side?
A line crossing both parallels: the alternate interior angle equals 35°, so it makes 35° with CD.

✏️ Practice — try these, take hints as needed

1. A transversal cuts two parallel lines; alternate interior angles are (2x + 25)° and (3x − 5)°. Find x.
Alternate interior angles are equal.
2x + 25 = 3x − 5.
Solve for x.
30
2. Co-interior angles on two parallel lines are (x + 40)° and (2x + 20)°. Find the smaller angle.
Co-interior angles sum to 180°.
(x+40)+(2x+20) = 180.
3x + 60 = 180 ⇒ x = 40.
80°
3. One angle a transversal makes with a parallel line is 118°. Find its corresponding angle at the other line.
Corresponding angles are equal.
No subtraction needed.
Copy the value.
118°
4. Two parallel lines cut by a transversal give co-interior angles in ratio 4 : 5. Find the smaller angle.
They sum to 180°.
4k + 5k = 180.
k = 20, smaller = 4k.
80°
5. A transversal makes an angle of 72° with the first of two parallel lines. Find the alternate exterior angle at the second line.
Alternate exterior angles are equal on parallels.
No supplement needed.
Same value.
72°

📝 Topic test — 8 questions

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