Alternate angles
Alternate angles x and 64°
x = 64°
Reason: alternate angles are equal
Angles, parallel lines and polygons How can one angle tell you all the others?
Foundation and Higher · AQA · 45 minutes
Students find eight angles from one, then learn to name each pair and write the reason that earns the mark. Polygons follow: triangles inside, one full turn outside. Slides to present, a vote that shows who's stuck, practice and homework.
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Walk once round a regular polygon with n sides and you turn 360° in all, so each exterior angle is 360° ÷ n. Move n from 3 to 12 (only whole n are polygons). For which n is the exterior angle a whole number of degrees? Could a regular polygon have an exterior angle of 50°?
They cross at (6, 60)
The idea in one minute
Angles on a straight line add up to 180°, angles round a point to 360°, and vertically opposite angles are equal. Where a line crosses two parallel lines, corresponding and alternate angles are equal and co-interior angles add up to 180°. A polygon with n sides splits into n − 2 triangles, so its angles add up to (n − 2) × 180°; its exterior angles always add up to 360°.
The lesson starts with one angle on a pair of parallel lines and asks for the other seven. Students then name each pair by where it sits (between the lines or not, same side or not), not only by letter shapes, and write each reason as the sentence a mark scheme looks for. Polygons come last, from triangles and from walking round the shape. It is Foundation content that Higher papers use too.
By the end, students can
- find angles on a line, round a point, vertically opposite and in a triangle
- name corresponding, alternate and co-interior angles and use them
- write the reason for each step as a full sentence
- find the angle sum of any polygon, and each angle of a regular one
Key words
- vertically opposite
- parallel, transversal
- corresponding, alternate, co-interior
- interior angle, exterior angle
- regular polygon
Before this lesson
Students should know that a right angle is 90°, add and take away angles, and solve a simple equation such as 3x + 30 = 180. The angle facts are revised here.
Next: Circle theorems.
The lesson, step by step
30 minutes in total. Each step says what changes for this length.
3 min 1 · Hook · students think
One angle, eight angles
Students find as many angles as they can from one, before anyone explains.
For this length: Show all eight angles at once; skip the hunt
Say“Two parallel lines, one line across them, and one angle given. How many of the other seven can you find? Write a reason beside each.”
Ask- Which angle did you find first, and how?
- Which ones are you sure of, and which are you guessing?
Watch forReading the size off the picture: ask for the reason, not what it looks like.
For the teacher
Show the slide: the lines y = 3 and y = −1 are parallel, and y = x + 1 crosses both. At the upper crossing the top-right angle is 45°.
Give two minutes alone. Most find the angle beside it (180° − 45°) and the one opposite it at once. Fewer reach the lower crossing: that is what parallel lines give.
Gather answers with Ask the class. Keep the reasons students gave: the next step names them properly.
Shorter: Show all eight angles at once; skip the hunt (−2 min).
9 min 2 · Explain · you explain
The facts, and the reasons to write
Four angle facts, the three parallel-line pairs, then polygons from triangles and one full turn.
For this length: Skip the four-facts slide: say the facts while naming the pairs
Say“Every angle you find needs its reason, written as a full sentence. A number with no reason loses marks.”
Ask- Are these two angles between the parallel lines or outside them?
- Are they on the same side of the line crossing them?
- How many triangles does a hexagon split into from one corner?
Watch forCalling co-interior angles equal: they are between the lines on the same side, and add up to 180°.
The angles of a hexagon
6 sides: 6 − 2 = 4 triangles
4 × 180° = 720°
Regular: each angle 720 ÷ 6 = 120°
For the teacher
First slide: the four facts students know. Say each as the sentence they will write: angles on a straight line add up to 180°; angles round a point add up to 360°; vertically opposite angles are equal; angles in a triangle add up to 180°.
Second slide: the parallel lines again. Name the four angles at each crossing top-left, top-right, bottom-left, bottom-right. Corresponding angles sit in the same place at each crossing. Alternate angles are between the lines, on opposite sides of the transversal. Co-interior angles are between the lines, on the same side. The letters F, Z and C help to spot them, but are never the reason.
Third slide: the reasons that earn the mark, word for word.
Fourth slide: a pentagon split into three triangles from one corner, so 540°. Any polygon: (n − 2) × 180°.
Fifth slide: walk round a polygon and you turn once, 360°. In a regular polygon each exterior angle is 360° ÷ n, and interior + exterior = 180°.
On the parallel-lines slide, point to each pair and ask the two questions (between? same side?) before naming it. On the last slide, move n and ask for a prediction first.
Shorter: Skip the four-facts slide: say the facts while naming the pairs (−4 min).
6 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves, with its reason.
For this length: Do the first two pairs only
Say“Watch me do one, with its reason. Then you do its twin, reason and all.”
Ask- Which fact are you using? Say the sentence before the number.
Watch forA right number with no reason, or the reason "Z angles".
Parallel lines
Co-interior angles are x and 110°. Find x.
x + 110 = 180, so x = 70°
Co-interior angles add up to 180°
Then theirs
Co-interior angles are x and 2x. Find x.
A triangle
A triangle's angles are x, 2x and 3x. Find x.
6x = 180, so x = 30°
Angles in a triangle add up to 180°
Then theirs
A triangle's angles are x, x + 20 and 2x. Find x.
A regular polygon
Each exterior angle is 24°. How many sides?
360 ÷ 24 = 15 sides
Exterior angles add up to 360°
Then theirs
Each interior angle is 150°. How many sides?
For the teacher
Work the first example aloud: the equation, the answer, then the reason as a sentence. Students solve its twin while you look at who writes a reason.
The polygon twin gives the interior angle: students need the exterior angle first (180° take away the interior angle).
Open the answers slide once most have finished their twins. Mark the reasons as well as the numbers.
Shorter: Do the first two pairs only (−3 min).
7 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
For this length: After the vote, the first six questions only
Say“Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- Where are the two angles: between the lines? On the same side?
Watch forChoosing by how the angles look rather than by where they sit.
Give each angle as a number of degrees. Write your reasons on paper as you go. The stretch is optional.
The question that shows who's stuck
- Two parallel lines are crossed by a third line. Two angles sit between the parallel lines, on the same side of the third line. One is 65°. What is the other?
In your Classroom, as they answer (an example) - A3
- B10
- C11
- D3
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- Three angles on a straight line are 40°, 75° and x. Find x.
- Four angles meet at a point: 90°, 110°, 85° and y. Find y.
- An isosceles triangle has an angle of 40° between its two equal sides. What size is each of its other two angles?
- Two horizontal parallel lines are crossed by a line sloping up to the right. At each crossing there are four angles: top-left, top-right, bottom-left and bottom-right. Sketch it first and mark the four at each crossing. Then put each pair of angles in the group that names it.They sort 10 items into Vertically opposite (equal) and Corresponding (equal) and Alternate (equal) and Co-interior (add up to 180°) and On a straight line (add up to 180°).
- Two angles sit between two parallel lines, on opposite sides of the line crossing them. One is 64°, so the other is 64° too. Which reason, written beside the working, earns the mark?
- Two co-interior angles are 2x + 10 and x + 20 (in degrees). They add up to 180°. Solve the equation line by line to find x. Write the numbers without the ° sign.They solve 2x + 10 + x + 20 = 180 line by line, each line checked as they write it.
- A regular decagon has 10 sides. Find the size of each interior angle.
- A pentagon's interior angles are x, x, 2x, 2x and 120°. Find x.
- Can a regular polygon have an exterior angle of 50°?
- Stretch: each interior angle of a regular polygon is 4 times its exterior angle. How many sides does it have?
Support · For the sorting question: for each pair, ask first whether both angles are at the same crossing. If not, ask whether they are between the parallel lines, then whether they are on the same side of the line crossing them.
For the teacher
Everyone chooses before anyone explains. Then ask a student for each answer to say where the two angles sit and which fact links them.
The first questions build fluency with the basic facts; then a sort of angle pairs, a reason to judge, an equation solved line by line, and polygons. The last ones ask for reasons and a stretch.
Students show their working and reasons on paper or the board; the quiz keeps their final answers.
If the vote splits, ask two students who chose differently to explain, then mark the two angles on the slide's lines together before anyone practises.
Shorter: After the vote, the first six questions only (−6 min).
Longer: The stretch, then: which regular polygons have an interior angle that is a whole number of degrees? (+10 min).
5 min 5 · Exit · students show
Angles with reasons
Two angles found and a reason written for each show who can do both.
Say“Find both angles, and write a reason for each step as a full sentence.”
Watch forTwo right numbers with "because they are the same" as the reason.
Find both angles and write a reason for each, as a full sentence.
- Two parallel lines are crossed by a third line. Angle p and an angle of 72° are alternate angles. Angle q is on a straight line with angle p. Find p and q, and write a reason for each.
For the teacher
Students answer the exit question on their own: two angles, a reason for each.
Then two or three read their reasons aloud; the class says whether each would earn the mark.
Read the reasons, not only the numbers: start the next lesson with two that would lose the mark, and ask the class to fix them. Set the homework.
Longer: Why do the exterior angles of any polygon add up to 360°? Explain to a partner (+5 min).
One to one: ask the student to say the reason aloud before writing any number, and to point to the two angles it links. Cover the diagram's numbers and ask which angles they could find from one.
Homework: find them, give the reasons
Find each angle or letter, and write a reason for every step as a full sentence (for example, "alternate angles are equal").
- Two angles on a straight line are 2x and 3x. Find x.
- A triangle has angles 40°, x and 2x + 20. Find x.
- Two corresponding angles on parallel lines are 4x − 30 and 2x + 40. Find x, then the size of each angle.
- Two co-interior angles are x + 15 and 2x + 15. Find x.
- Find the sum of the interior angles of an octagon (8 sides). Then find each interior angle of a regular octagon.
- Each interior angle of a regular polygon is 162°. How many sides does it have?
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching angles, parallel lines and polygons: common questions
How can students tell alternate, corresponding and co-interior angles apart?
Ask two questions about the pair. Are both angles between the parallel lines? Are they on the same side of the line crossing them? Between and opposite sides: alternate (equal). Between and the same side: co-interior (they add up to 180°). The same place at each crossing: corresponding (equal). The letters Z, C and F help to spot them, but are never the written reason.
What reasons do GCSE mark schemes accept for angles?
The name of the fact and the fact itself, as a sentence: "alternate angles are equal", "co-interior angles add up to 180°", "angles on a straight line add up to 180°". Shapes ("Z angles"), the sum on its own, or "they look the same" do not earn the reason mark. The lesson practises the sentences on a slide, in a quiz question and in the exit.
How do you find each interior angle of a regular polygon?
Find the exterior angle first: 360° ÷ the number of sides. The interior angle is 180° minus that. For a regular nonagon (9 sides), 360 ÷ 9 = 40°, so each interior angle is 140°. Going the other way, from an interior angle to the number of sides, find the exterior angle and divide it into 360°.
Is this angles lesson for Foundation or Higher GCSE?
Both. The facts are Foundation content, and Higher papers use them inside longer problems and proofs. Foundation groups may stop after the regular polygon questions; the stretch and the extra (which regular polygons have whole-number interior angles) suit Higher.
Can I use the lesson without creating an account?
Yes. Read the lesson, print the worksheet and open its slides without signing in; nothing you draw there is kept. A free teacher account keeps the whole lesson in your Library to change and teach, with the quizzes, the answers and the answer slides.
Curriculum reference: AQA GCSE Mathematics 8300: G1 (conventional terms and notation: parallel lines, polygons) and G3 (angles at a point, on a line and vertically opposite; alternate and corresponding angles on parallel lines; the angle sum of a triangle, used for any polygon and for regular polygons), with G4 (names of polygons and triangles) and G6 (the base angles of an isosceles triangle are equal).
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