Parallel lines cut by a transversal: the eight angles and the angle pairs (corresponding, alternate interior, alternate exterior, co-interior or same-side interior, vertically opposite, linear pairs), which are equal and which add to 180°, the converses that prove lines parallel, the perpendicular transversal theorems, worked examples with algebra, the mistakes, and a printable chart with practice

"Parallel lines cut by a transversal" is searched about 5,100 times a month in the United States, almost entirely by Grade 8 and Geometry students facing a diagram with eight angles and a list of names to attach to them, and the UK, Australian and Indian versions of the same question (alternate angles, co-interior angles, "Z angles") arrive at the same page. The content is small: two angle sizes and five names. This page gives the chart with every angle labelled, the names and the shapes that go with them, which are equal and which add to 180°, the converses for proving lines parallel, the perpendicular transversal theorems, worked examples including the algebraic kind, the mistakes, and a printable chart with practice.

Download the angle pairs chart with practice (PDF) (A4, the labelled figure, the pairs table and the theorems on page 1, the worked examples and the practice set with answers on page 2, free).

The figure: eight angles, two sizes

Draw two parallel lines, l and m, and a transversal t crossing both. At each crossing there are four angles, eight in all, numbered 1 to 4 at the top crossing (1 top left, 2 top right, 3 bottom left, 4 bottom right) and 5 to 8 at the bottom crossing in the same positions.

The whole topic rests on one fact: when the lines are parallel, the eight angles have only two sizes. Every acute angle equals every other acute angle; every obtuse angle equals every other obtuse angle; any acute angle plus any obtuse angle is 180°. (If the transversal is perpendicular, all eight are 90°.) Once you know one angle you know all eight. The names exist so that you can say why two particular angles are equal.

Interior angles are the four between the parallel lines (3, 4, 5, 6); exterior angles are the four outside (1, 2, 7, 8).

The angle pairs

The names vary by country, and the mark scheme accepts its own: England says corresponding, alternate, co-interior (and children say F, Z and C angles, though examiners want the words); the US says corresponding, alternate interior, alternate exterior, same-side interior (or consecutive interior); India uses corresponding, alternate interior, alternate exterior, co-interior. Vertically opposite angles are "vertical angles" in the US.

The converses: proving lines parallel

Each relationship runs backwards. If a transversal crosses two lines and

• a pair of corresponding angles is equal, or
• a pair of alternate interior (or alternate exterior) angles is equal, or
• a pair of co-interior angles adds to 180°,

then the two lines are parallel. One pair is enough. This is how you prove parallelism in a diagram: measure or calculate one pair, name the pair, state the converse. Vertically opposite angles and linear pairs tell you nothing about parallelism, since they hold at any crossing.

The perpendicular transversal theorems

• If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other as well. (All eight angles are 90°: corresponding angles equal.)
• In a plane, two lines perpendicular to the same line are parallel to each other. (Corresponding angles are both 90°, so the converse applies.)
• In a plane, two lines parallel to the same line are parallel to each other (the transitivity of parallel lines).

These three are the "perpendicular transversal theorem" and its companions in US Geometry; the UK and Indian syllabuses use them without naming them.

Six worked examples

1. One angle given. Angle 1 = 65°. Then 4 = 65° (vertically opposite), 2 = 3 = 115° (linear pairs), 5 = 65° (corresponding to 1), 8 = 65°, 6 = 7 = 115°. Two sizes, done.
2. Alternate interior with algebra. Angles 3 and 6 are alternate interior, measuring (2x + 15)° and (3x − 20)°. Equal: 2x + 15 = 3x − 20, so x = 35; each angle is 85°.
3. Corresponding with algebra. (3x + 10)° and (5x − 30)° are corresponding. 3x + 10 = 5x − 30, so 2x = 40, x = 20; each is 70°.
4. Co-interior with algebra. (4x)° and (2x + 30)° are co-interior. They add to 180°: 6x + 30 = 180, x = 25; the angles are 100° and 80°.
5. Proving parallel. Two lines are crossed by a transversal; the alternate interior angles measure 72° and 72°. Equal alternate interior angles, so by the converse the lines are parallel.
6. Not parallel. Corresponding angles measure 68° and 71°. Not equal, so the lines are not parallel; they meet on the side where the angles are smaller.

The mistakes that cost marks

• Calling co-interior angles equal. They add to 180°; the equal pairs are corresponding and alternate.
• Using the parallel-line facts when the lines are not marked parallel. Only vertically opposite angles and linear pairs hold for any two lines.
• Writing "Z angles" or "F angles" in an exam answer. Write alternate or corresponding; the letters are a memory aid, not a reason.
• Mixing up alternate interior (Z, opposite sides) with co-interior (C, same side).
• Forgetting that the two crossings each have their own four angles: angle 2 and angle 6 are corresponding, but angle 2 and angle 5 are not a named pair at all (they are supplementary, by two steps).
• Solving algebra pairs as equal when they are supplementary, or the reverse. Identify the pair first, then write the equation.

Where it sits in each curriculum

Practice (answers in the PDF)

1. Angle 2 in the figure is 112°. Find angles 3, 5 and 8, and name the pair each makes with angle 2 or another angle you used.
2. Angles (5x − 10)° and (3x + 30)° are alternate interior angles between parallel lines. Find x and the angles.
3. Angles (7x)° and (2x + 45)° are co-interior. Find x and both angles.
4. A transversal makes a 90° angle with one of two parallel lines. What angle does it make with the other, and which theorem says so?
5. Two lines are crossed by a transversal and the co-interior angles are 95° and 85°. Are the lines parallel? Why?
6. In a diagram, two angles are corresponding and measure (4x + 8)° and (6x − 22)°. Find x. Are the lines parallel for that x? Explain.

Download the chart and the practice set (PDF)

Where to get help

A 1-on-1 online maths class with a tutor who holds a master's in mathematics and teaches Grade 8 and Geometry, KS3 and GCSE, the Australian Year 7 to 10 and CBSE Class 7 and 9 every week: angle facts taught on a shared diagram with every reason written in words, so "alternate angles are equal" becomes a sentence the child writes without prompting. The Geometry page lists the units; the free 30-minute first class is a real lesson on this week's homework. Related pages: the polygon names chart with angle sums and the trapezium.

Questions parents ask

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