GCSE · Maths · Edexcel · Spec 1MA1 · Foundation+Higher

Angle properties and parallel lines

Tilt one line by a single degree and three angle 'rules' break at once. By the end you'll know why, and why no regular polygon has a 161° angle.

Parallel lines · live figure

Tilt one line and watch the rules break
56°56°124°56°ABEFGCH

∠AGE 56°. ∠BGH 56°. ∠AGH 124°. ∠CHG 56°. Classification: EF is the transversal: it crosses both lines. Relationship: Only when CH is parallel to AB: ∠AGE = ∠CHG (corresponding), ∠BGH = ∠CHG (alternate) and ∠AGH + ∠CHG = 180° (co-interior).

∠AGE56°∠BGH56°∠AGH124°∠CHG56°drag H up or down the transversal

EF is the transversal: it crosses both lines

Only when CH is parallel to AB: ∠AGE = ∠CHG (corresponding), ∠BGH = ∠CHG (alternate) and ∠AGH + ∠CHG = 180° (co-interior).

Line AB is fixed. Line CH is hinged at C, and its end H slides along the transversal EF. Drag H. While CH sits on the faint guide it is parallel to AB. Tilt it off the guide and compare the four readings.

Watch out: Co-interior angles are NOT equal. Look at ∠AGH and ∠CHG: on parallel lines they add up to 180°. Tilt CH and even that stops working.

Name the pair

Which kind of pair is it?

Pick a pair, then choose its type. Find each angle on the figure first: the middle letter tells you which crossing it is at.

Still to sort

Corresponding (0)

Same corner at each crossing

Where the line is: Corresponding pairs use one angle between the lines and one outside them. Alternate pairs are both between (or both outside).

Alternate (0)

Opposite corners, opposite sides of the transversal

Where the line is: Alternate and co-interior are both between the lines. Alternate: opposite sides of the transversal. Co-interior: the same side.

Co-interior (0)

Same side of the transversal, both between the lines

Where the line is: Co-interior angles are not equal on parallel lines. They add up to 180°.

Vertically opposite (0)

Opposite each other at ONE crossing

Where the line is: Vertically opposite angles share a vertex. Every other pair here uses two different crossings, G and H.

7 of 7 still to sort.

Use the figure at the top of the lesson, with CH on its guide. In three-letter names the middle letter is the vertex, so ∠AGE is the angle at G between GA and GE.

Watch out: The name of a pair comes from where the angles sit, not from their sizes. Corresponding angles are still corresponding when the lines are not parallel. They just stop being equal.

Where the rules come from

Every angle fact comes from two turns: a full turn is 360° and a half turn is 180°.

Points A, B and C are drawn with B in the middle, and ABC looks perfectly straight. At B, three angles sit side by side between BA and BC: 73°, 90° and 19°. Do A, B and C lie on a straight line?

Why the facts are true

Problem

Prove two facts using only angles on a straight line and alternate angles: (1) vertically opposite angles are equal; (2) the angles in any triangle add up to 180°.

Spot the slip

One reason is wrong

Lines AB and CD are parallel. The line PQ crosses AB at P and CD at Q, and B and D are on the same side of PQ. Angle BPQ = 105°. R is a point on CD on the D side of Q, and triangle PQR is isosceles with QP = QR. Find angle QPR, giving a reason for each step.

A student's answer — which line goes wrong?

Polygons, forwards and backwards

Fill the gaps in the method

(a) The interior angles of a polygon add up to 3240°. How many sides does it have? (b) Can a regular polygon have interior angles of 160°? What about 161°?

  1. Pick one vertex and draw a line segment to every other vertex. A polygon with n sides always splits into n − 2 triangles.Try it: a pentagon (5 sides) splits into 3 triangles, and a hexagon (6 sides) into 4. Always two fewer.
  2. missing step
Which line is step 2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A full turn is 360° and a half turn is 180°. Every other angle fact follows from those two.

What you need to know

  • An angle is a measure of turn, in degrees: a full turn is 360° and a half turn is 180°.
  • Angles round a point add up to 360°. Adjacent angles on a straight line add up to 180°.
  • Vertically opposite angles are equal.
  • A transversal is a line that crosses two or more lines at different points.
  • On parallel lines, corresponding angles are equal, alternate angles are equal and co-interior angles add up to 180°.
  • The angles in a triangle add up to 180°. A polygon with n sides has interior angles adding up to 180(n − 2).

The big picture

An angle measures turn. A full turn is 360° and a half turn is 180°, so angles round a point add up to 360° and adjacent angles on a straight line add up to 180°. Vertically opposite angles are equal. When a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal and co-interior angles add up to 180°. The same facts test whether lines are parallel or points are in a straight line. Together they prove that a triangle's angles add up to 180°, and a polygon with n sides has interior angles adding up to 180(n − 2).

Key points

1Two angles that add up to 180° are supplementary. Only ADJACENT angles on a straight line are guaranteed to add up to 180°.
2To check three points are in a straight line, add the adjacent angles at the middle point. It's a straight line only if they make exactly 180°.
3Name angles with three letters. The middle letter is the vertex, and the outer two letters can go either way round.
4On parallel lines every angle is either x or 180° − x.
5Lines are parallel if corresponding angles are equal, alternate angles are equal or co-interior angles add up to 180°. If the fact fails, they are not parallel. Angles at one crossing alone can never show this.
6An isosceles triangle has two equal angles. Every angle of an equilateral triangle is 180 ÷ 3 = 60°.
7Each interior angle of a regular polygon is 180(n − 2) ÷ n. It exists only if solving for n gives a whole number.

Worked example

Problem

A hexagon has interior angles of 128°, 105°, 140°, 117°, 96° and x. Work out x.

⚠ Watch out

Writing that co-interior angles are equal. Alternate and corresponding angles are equal, but co-interior angles (same side, both between the lines) add up to 180°. If one is 70°, the other is 110°, not 70°.

🧠

Memory hook

Two turns, two sizes. A full turn is 360° and a half turn is 180°. On parallel lines there are only two sizes, x and 180 − x: matching pairs are equal, and a same-side inside pair shares a half turn.

✓

Check yourself

A transversal cuts two lines, making co-interior angles of 58° and 121°. Are the lines parallel? (No: 58 + 121 = 179°, but on parallel lines co-interior angles make exactly 180°.)

Flashcards

(15)
Angles round a point add up to…?
360°, a full turn. For example m + 148 = 360 gives m = 212°.
Adjacent angles on a straight line add up to…?
180°, a half turn. Two angles that add up to 180° are called supplementary.
Two lines cross. What do you know about the opposite angles?
Vertically opposite angles are equal.
What is a transversal?
A line that crosses two or more lines at different points.
In ∠PQR, which point is the vertex?
Q, the middle letter. ∠PQR and ∠RQP are the same angle.
Two angles sit in the matching corner at two different crossings. What is the pair called, and when are they equal?
Corresponding angles. They are equal when the lines are parallel.
Both between the lines, on opposite sides of the transversal: which pair is that?
Alternate angles, which are equal on parallel lines. A pair both OUTSIDE the lines on opposite sides is alternate too.
Which pair of angles on parallel lines is NOT equal, and what do they do instead?
Co-interior angles: same side of the transversal, both between the lines. They add up to 180°.
How can you prove two lines are NOT parallel?
Show that a corresponding or alternate pair is unequal, or a co-interior pair doesn't add up to 180°.
How do you check whether three points lie on a straight line?
Add the adjacent angles at the middle point. It's a straight line only if they make exactly 180°.
Why do the angles in a triangle add up to 180°?
Draw a line through one vertex parallel to the opposite side. Two pairs of alternate angles are equal, and the three angles at that vertex lie on a straight line.
Isosceles and equilateral triangles: what do you know about their angles?
Isosceles: two angles are equal. Equilateral: all three are equal, so each is 180 ÷ 3 = 60°.
Interior angle sum of a polygon with n sides?
180(n − 2), because it splits into n − 2 triangles from one vertex. For example a pentagon has 540°.
Each interior angle of a regular polygon with n sides?
180(n − 2) ÷ n. For example a regular heptagon has 900 ÷ 7 ≈ 128.6°.
How do you decide whether a regular polygon can have a given interior angle?
Solve 180(n − 2) ÷ n = angle for n. It's possible only if n is a whole number.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

More Edexcel GCSE Maths topics

How this lesson was checked. This Edexcel GCSE Maths (specification 1MA1)lesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 29 September 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.