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Parallel lines and a transversal

UPCAT Math Review · Geometry

Parallel Lines and a Transversal

Eight angles, but only two different sizes.

Two parallel lines are cut by a transversal

One angle is 65°. How big is its co-interior partner?

Number the eight angles.

Four at the top crossing, four at the bottom one.

Say angle 1 is the 65° one.

The little arrowheads on the two horizontal lines are the standard mark for “these are parallel”. Without them, none of the rules below apply.

At one crossing: only two sizes.

vertical angles

across the X — equal

∠1 = ∠4 = 65°

linear pair

side by side — add to 180°

∠2 = 180 − 65 = 115°

So the top crossing is 65, 115, 115, 65 — and that is already the whole story.

Carry it down to the second crossing.

Because the lines are parallel, the bottom crossing is a copy of the top one.

Pair name	Where they sit	Rule
corresponding	same corner, both crossings	equal
alternate interior	inside, opposite sides	equal
alternate exterior	outside, opposite sides	equal
co-interior	inside, same side	add to 180°

Three of the four say “equal”. Only the same-side pair adds to 180°.

Names are worth knowing for exams, but the shortcut is simpler: every angle is either 65° or 115°, and nothing else is possible.

Fill them all in.

Angle	Size	Why
∠1	65°	given
∠2	115°	linear pair with ∠1
∠3	115°	linear pair with ∠1
∠4	65°	vertical to ∠1
∠5	65°	corresponding to ∠1
∠6	115°	linear pair with ∠5
∠7	115°	linear pair with ∠5
∠8	65°	vertical to ∠5

four angles of 65°, four of 115°

65 + 115 = 180, so any two different-sized angles here are supplementary.

The shortcut, and when it fails.

same look

both acute, or both obtuse

they are equal

different look

one acute, one obtuse

they add to 180°

Glance at the diagram, decide whether the two angles look the same or opposite, and you have the relationship without naming it.

Only when the lines are parallel

Remove the arrowheads and every rule in Step 3 collapses. Vertical angles and linear pairs still hold — those never needed parallel lines.

Variations: expressions that are equal, expressions that total 180, lines that are not parallel.

p ∥ q. Alternate interior angles measure (7x − 18)° and (4x + 27)°. Find the angle.

7x − 18 = 4x + 27

alternate interior: equal

3x = 45, x = 15

− 4x, + 18, ÷ 3

7(15) − 18 = 87°

4(15) + 27 = 87 ✓

Different: x is not the angle. Both expressions must give the same number when we substitute.

p ∥ q. Co-interior (same-side) angles measure (3x + 14)° and (5x − 2)°. Find both.

(3x + 14) + (5x − 2) = 180

same side: total 180

8x + 12 = 180, x = 21

− 12, ÷ 8

77° and 103°

one acute, one obtuse ✓

Equal or 180? Decide before writing

Setting the co-interior pair equal gives 3x + 14 = 5x − 2, so x = 8 and both angles 38°. Two acute angles cannot sit on the same side between parallels — they would have to total 180.

A transversal crosses lines p and q. Co-interior angles measure 98° and 84°. Are p and q parallel?

98 + 84 = 182

add the same-side pair

182 ≠ 180

the rule fails

not parallel

extended far enough, they meet

Different: the rules run backwards too. Equal corresponding angles, or a same-side pair totalling exactly 180, prove the lines parallel; anything else proves they are not.

p ∥ q. One angle is 125°. Its linear-pair partner is (x + y)° and its corresponding angle is (x + 3y)°. Find x and y.

x + y = 55

linear pair: 180 − 125

x + 3y = 125

corresponding: equal

2y = 70, y = 35, x = 20

subtract the first from the second

Different: two unknowns need two equations — take one from each angle the figure ties to the 125°.

The zigzag: draw a third parallel line.

p ∥ q. A bent path runs from A on p to a point B between the lines, then to C on q. The angles at A and C are 35° and 50°. Find x, the angle at B.

dashed line through B, parallel to p and q

it splits x in two

upper part = 35°

alternate interior with A

lower part = 50°

alternate interior with C

x = 35 + 50 = 85°

add the two pieces

Measure the outer angles on the inside

The rule “x = sum” needs the angles at A and C measured between the parallel line and the path, on the side the bend points to. If the figure gives the outside angle (here 145° instead of 35°), take 180 minus it first.

A triangle between the parallels. If A is on p and B, C are on q, the angles at A on either side of the triangle are alternate interior copies of ∠B and ∠C. All three angles at A lie on line p, so with ∠B = 55° and ∠C = 64° the angle of the triangle at A is 180 − 55 − 64 = 61°.

On the exam

Expect the angles as expressions — set them equal (corresponding, alternate) or make them total 180 (same-side) — then a final step to a neighbouring angle, or a bend that needs the extra parallel line. Decide “equal or 180?” by the look: both acute or both obtuse means equal. Ten-second check: along one transversal every angle is one of only two sizes, and those two add to 180°.

Check: the sums must work.

around a point 65 + 115 + 65 + 115 = 360 one full turn ✓

co-interior ∠4 + ∠6 = 65 + 115 = 180 ✓

corresponding ∠1 = ∠5 = 65 equal ✓

Remember the flow

One angle · Two sizes · 65 or 115

Same look, equal. Different look, 180.

Now you try

Practice problem

Find x

One co-interior pair is (3x + 10)° and (2x − 5)°. Find x.

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