Rays and angle measures
A line goes on forever both ways. A line segment has two end points. A ray has one end point and goes on forever in one direction, like a torch beam.
Two rays from the same point make an angle. The shared point is the vertex, the rays are the arms. We measure the turn in degrees; a full turn is 360°.
| Type | Size |
|---|---|
| Acute | more than 0°, less than 90° |
| Right | exactly 90° |
| Obtuse | more than 90°, less than 180° |
| Straight | exactly 180° |
| Reflex | more than 180°, less than 360° |
Two angles that add to 90° are complementary; two that add to 180° are supplementary. Adjacent angles share a vertex and an arm, and do not overlap.
Linear pair and vertically opposite angles
Linear pair: if a ray stands on a line, the two adjacent angles it makes add up to 180°. This is the linear pair axiom. The reverse is also accepted: if two adjacent angles add to 180°, their outer arms form a straight line.
Vertically opposite angles are equal
When two lines cross, they make four angles. Name them ∠1, ∠2, ∠3, ∠4 going round.
- ∠1 + ∠2 = 180° (linear pair on one line)
- ∠2 + ∠3 = 180° (linear pair on the other line)
- So ∠1 + ∠2 = ∠2 + ∠3. Take ∠2 away from both sides: ∠1 = ∠3.
In the same way ∠2 = ∠4. This is a proper proof: every step follows from something already accepted.
Parallel lines and transversals
Parallel lines lie in one plane and never meet; we write l ∥ m. A transversal is a line that cuts two or more lines at different points. It makes 8 angles: 4 at each crossing.
- Corresponding angles (F shape): same position at each crossing. If l ∥ m, they are equal.
- Alternate interior angles (Z shape): inside the two lines, on opposite sides of the transversal. If l ∥ m, they are equal.
- Interior angles on the same side (co-interior, C shape): if l ∥ m, they add up to 180°.
The converse
If a transversal makes equal corresponding angles (or equal alternate angles, or co-interior angles adding to 180°), then the two lines are parallel. Also, two lines parallel to the same line are parallel to each other.
Proof by contradiction
Sometimes the easiest way to prove a statement is to suppose it is false and show this leads to something impossible (a contradiction). Then the statement must be true.
Example: A transversal makes co-interior angles that add to 180°. Prove l and m cannot meet.
- Suppose l and m do meet, at a point P. Then l, m and the transversal make a triangle.
- Two angles of this triangle are the co-interior angles, which already add to 180°.
- The third angle at P must be more than 0°, so the triangle's angles add to more than 180°.
- But a triangle's angles always add to exactly 180°. Contradiction!
- So our supposition was wrong: l and m never meet. They are parallel.
Steps to remember: assume the opposite → reason carefully → reach something impossible → so the original statement is true.
Try it yourself (practical)
At home: Open a pair of scissors. The blades make an X. Mark the four angles on paper under them with a pencil and measure with a protractor: opposite angles come out equal every time.
Notebook test: Draw a slanting line across two ruled lines of your notebook. Measure the top-right angle at each crossing. They match, because ruled lines are parallel.
In the 3D: go to the last step. Predict what happens to the bottom angle if you tilt m by 10°. Then try it.
Key formulas and definitions
- Linear pair: ∠a + ∠b = 180°
- Vertically opposite: ∠1 = ∠3, ∠2 = ∠4
- Angles round a point = 360°
- If l ∥ m: corresponding angles equal (F)
- If l ∥ m: alternate interior angles equal (Z)
- If l ∥ m: co-interior angles add to 180° (C)
Worked examples
1. One angle of a linear pair is 47°. Find the other.
The two add to 180°. Other angle = 180° − 47° = 133°.
2. Two lines cross. One angle is 72°. Find the other three.
Opposite angle = 72° (vertically opposite). Each neighbour = 180° − 72° = 108° (linear pair). So the angles are 72°, 108°, 72°, 108°.
3. Angles of a linear pair are in the ratio 2 : 3. Find them.
Let them be 2x and 3x. 2x + 3x = 180°, so 5x = 180°, x = 36°. Angles: 72° and 108°.
4. l ∥ m and a transversal makes a 65° angle at l (top right). Find the corresponding angle at m and a co-interior angle.
Corresponding angle at m = 65°. The bottom-left angle at l is also 65° (vertically opposite), and the interior angle at m on the same side of the transversal is 180° − 65° = 115°.
5. Two co-interior angles are (3x + 10)° and (2x − 5)°. The lines are parallel. Find x.
Co-interior angles add to 180°: 3x + 10 + 2x − 5 = 180, so 5x + 5 = 180, 5x = 175, x = 35. Angles: 115° and 65°.
6. AB ∥ CD. From a point E between them, ∠BAE = 40° and ∠DCE = 30°. Find ∠AEC.
Draw a line through E parallel to AB (and so to CD). It splits ∠AEC into two parts. By alternate angles, one part = 40° and the other part = 30°. ∠AEC = 40° + 30° = 70°.
7. Prove by contradiction: two different lines can meet in at most one point.
Suppose two different lines meet at two points P and Q. Then both lines pass through P and Q. But through two points there is only one line (Euclid's postulate). So the lines are the same — a contradiction. Hence they meet in at most one point.
Common mistakes
- Thinking vertically opposite angles add to 180°: they are equal; it is the neighbouring angles that add to 180°.
- Using 'alternate angles are equal' when the lines are not parallel: the rule needs l ∥ m.
- Mixing up co-interior and alternate angles: co-interior (C) add to 180°, alternate (Z) are equal.
- In proof by contradiction, forgetting to say which assumption was wrong at the end.