Cambridge Lower Secondary · Mathematics 0862 · Stage 8 · Geometry and Measure
Angles in Triangles and on Parallel Lines
Prove and use the exterior-angle fact for triangles, and find angles on intersecting and parallel lines using vertically opposite, corresponding and alternate angles — with reasons.
British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course
- Explain (derive) why the exterior angle of a triangle is equal to the sum of the two interior opposite angles, and use this fact to find missing angles. [8Gg.10]
- Recognise and describe the angle properties of parallel lines and intersecting lines, using words such as alternate, corresponding and vertically opposite. [8Gg.11]
In Stage 8 you stop just measuring angles and start reasoning about them. You use a small set of angle facts to work out angles you cannot measure, and you give a reason for every step. This page teaches the facts for intersecting lines, parallel lines and triangles, shows you how to prove the exterior-angle fact, and practises the 'find the angle, give a reason' questions that appear in every Checkpoint-style test.
1. Key words
| Word / phrase | Meaning / use |
|---|---|
| intersecting lines | خطوط متقاطعة — lines that cross at a point |
| vertically opposite angles | زاويتان متقابلتان بالرأس — the angles opposite each other where two lines cross; they are equal |
| parallel lines | خطوط متوازية — lines that never meet and are always the same distance apart; shown with matching arrows |
| transversal | القاطع — a line that crosses two (or more) other lines |
| corresponding angles | زاويتان متناظرتان — angles in the same position at each crossing (F shape); equal on parallel lines |
| alternate angles | زاويتان متبادلتان — angles between the parallel lines on opposite sides of the transversal (Z shape); equal |
| co-interior angles | زاويتان داخليتان في جهة واحدة — between the parallel lines on the same side (C shape); add up to 180° |
| interior angle | زاوية داخلية — an angle inside a shape |
| exterior angle | زاوية خارجية — the angle between one side of a shape and the next side extended (made longer) |
| interior opposite angles | الزاويتان الداخليتان البعيدتان — the two angles inside the triangle that are not next to the exterior angle |
| derive / prove | يستنتج / يبرهن — show why a fact must be true using facts you already know |
2. Facts you already know
- Angles on a straight line add up to 180°.
- Angles around a point add up to 360°.
- Angles in a triangle add up to 180°.
- In an isosceles triangle the two base angles are equal; in an equilateral triangle every angle is 60°.
3. Vertically opposite angles
When two straight lines cross, they make four angles. The angles opposite each other are equal. Here is why: a + b = 180° (straight line) and b + (the angle opposite a) = 180° (another straight line), so the angle opposite a must equal a.
4. Angles on parallel lines
Parallel lines are marked with matching arrows (> or >>). When a transversal crosses two parallel lines, it makes the same angles at both crossings. This gives three pairs to learn:
| Pair | Where they are | Shape to look for | Fact |
|---|---|---|---|
| Corresponding | same position at each crossing (e.g. both above the line, to the right of the transversal) | F (it may be backwards or upside down) | equal |
| Alternate | between the parallel lines, on opposite sides of the transversal | Z (or N, or a backwards Z) | equal |
| Co-interior | between the parallel lines, on the same side of the transversal | C (or U) | add up to 180° |
| Vertically opposite | opposite each other at one crossing | X | equal |
Corresponding angles and alternate angles are equal only if the lines are parallel. If the diagram has no arrows and the question does not say the lines are parallel, you cannot use these facts. (Vertically opposite angles are always equal — they do not need parallel lines.)
- F = متناظرتان (corresponding) → متساويتين.
- Z = متبادلتان (alternate) → متساويتين.
- C = داخليتان في جهة واحدة (co-interior) → مجموعهم 180°.
- X = متقابلتان بالرأس (vertically opposite) → متساويتين.
- بس خلي بالك: F و Z و C تنفع بس لو الخطين متوازيين (عليهم أسهم).
5. The exterior angle of a triangle
Extend one side of a triangle (make it longer). The angle between the extended side and the next side is an exterior angle. In the diagram, d is the exterior angle at C, and a and b are the interior opposite angles (the two angles inside the triangle that are not next to d).
Derivation (proof) — learn these three lines:
- a + b + c = 180° (angles in a triangle add up to 180°).
- c + d = 180° (angles on a straight line add up to 180°).
- Both totals equal 180°, so a + b + c = c + d. Take c away from both sides: d = a + b.
The exterior angle of a triangle is equal to the sum of the two interior opposite angles. This saves a step: you do not need to find c first.
6. Why the angles in a triangle add up to 180° (using parallel lines)
Draw a line through A parallel to BC. The angle marked b at A is equal to angle B (alternate angles), and the angle marked c at A is equal to angle C (alternate angles). At A, the three angles b, a and c lie on a straight line, so b + a + c = 180°. These are exactly the three angles of the triangle. This is a good example of convincing — using known facts to prove a new one.
7. Giving reasons
Checkpoint questions often say 'Give a reason for your answer'. Use the correct angle words — 'Z angles' or 'F angles' may not get the mark. Learn these reasons:
| Fact | Reason to write |
|---|---|
| a + b = 180° | angles on a straight line add up to 180° |
| sum = 360° | angles around a point add up to 360° |
| equal (X) | vertically opposite angles are equal |
| equal (F) | corresponding angles are equal |
| equal (Z) | alternate angles are equal |
| sum = 180° (C) | co-interior angles add up to 180° |
| triangle | angles in a triangle add up to 180° |
| exterior angle | the exterior angle of a triangle equals the sum of the interior opposite angles |
| isosceles | base angles of an isosceles triangle are equal |
8. Worked examples
Worked example 1 — Work out the size of angle x. Give a reason for your answer. [2]
- x is the exterior angle; 67° and 48° are the interior opposite angles.
- x = 67° + 48° = 115°.
- Reason: the exterior angle of a triangle is equal to the sum of the two interior opposite angles.
Check another way: third angle = 180 − 67 − 48 = 65°, then x = 180 − 65 = 115° ✓
Worked example 2 — An exterior angle of a triangle is 130°. One of the interior opposite angles is 75°. Work out the other interior opposite angle. [1]
- 75 + ? = 130 (exterior angle = sum of interior opposite angles).
- ? = 130 − 75 = 55°.
Common mistake: doing 180 − 130 − 75. The 180 − 130 = 50° is the interior angle next to the exterior angle — it is not one of the opposite angles.
Worked example 3 — Two straight lines cross. One of the angles is 38°. Work out the other three angles. [2]
- The angle opposite 38° is 38° (vertically opposite angles are equal).
- The angles next to it are 180 − 38 = 142° each (angles on a straight line add up to 180°).
- Check: 38 + 142 + 38 + 142 = 360° (angles around a point) ✓
Worked example 4 — Use the diagram of the parallel lines in section 4 with a transversal at 60°. If the angle above the top line and to the right of the transversal is 60°, find (a) the angle in the same position at the bottom line, (b) the angle below the top line on the left of the transversal, (c) the angle below the top line on the right of the transversal. Give reasons. [3]
- (a) 60° — corresponding angles are equal.
- (b) 60° — vertically opposite to the 60° angle (it is also alternate to the 60° angle at the bottom line).
- (c) 180 − 60 = 120° — angles on a straight line add up to 180°. (It is co-interior with the 60° angle at the bottom line: 120 + 60 = 180 ✓)
Worked example 5 — PQ and RS are parallel. Work out angles p, q and r. Give a reason for each. [3]
- p = 52° (alternate angles are equal: angle PAB and angle ABC).
- q = 61° (alternate angles are equal: angle QAC and angle ACB).
- r = 180 − 52 − 61 = 67° (angles on a straight line at A add up to 180°; or angles in triangle ABC add up to 180°).
Worked example 6 — In a triangle, the two interior opposite angles are 2x and 3x. The exterior angle is 125°. Work out x. [2]
- Exterior angle = sum of interior opposite angles: 2x + 3x = 125.
- 5x = 125, so x = 25°.
- Check: 2 × 25 + 3 × 25 = 50 + 75 = 125 ✓
- Using alternate or corresponding angles when the lines are not parallel (look for the arrows).
- Mixing up alternate (equal) and co-interior (add to 180°). Ask: are the angles on the same side of the transversal or opposite sides?
- Adding the wrong angles for the exterior angle: use the two angles far away from it, not the one next to it.
- Writing 'Z angles' or 'F angles' as the reason — write 'alternate angles are equal' or 'corresponding angles are equal'.
- Measuring with a protractor when the diagram says NOT TO SCALE — you must calculate.
- Forgetting the ° sign, or giving a reflex angle when the question wants the angle inside the shape.
- 'Work out the size of angle x. Give a reason for your answer.' — one mark for the angle, one for the reason with correct vocabulary.
- 'Complete the sentence: angle a and angle b are ______ angles.' — write alternate, corresponding, co-interior or vertically opposite.
- Tick or ring the correct statement, e.g. 'a = b because they are corresponding / alternate / vertically opposite'.
- Multi-step diagrams: find one angle at a time, write each angle on the diagram as you go, and give a reason for each step.
- Show-that / prove questions: 'Show that the exterior angle equals …' — write the facts in order with reasons, as in section 5.
- Algebra in angles: form an equation (for example 2x + 3x = 125) and solve it.
Practice
- Vertically opposite angles are equal; angles on a straight line add up to 180°.
- On parallel lines: corresponding angles are equal (F), alternate angles are equal (Z), co-interior angles add up to 180° (C).
- Exterior angle of a triangle = sum of the two interior opposite angles (proof: triangle 180° and straight line 180°).
- Always give a reason in proper words: 'alternate angles are equal', not 'Z angles'.
- Check for parallel arrows before using F, Z or C facts.
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