Angles and constructions
Intersecting lines
Cross two lines and four angles appear. Two facts let you work out all of them from just one.
Opposite angles are equal
When two lines INTERSECT (cross), they make four angles. The angles directly across from each other are OPPOSITE ANGLES, and they are equal. If the four angles are a, b, c, d going round, then a = c and b = d.
Angles on a straight line
Angles that sit next to each other on a straight line add up to 180°. So if one angle at the crossing is 53°, the angle beside it is 180° − 53° = 127°. Put the two facts together and one known angle gives you all four: 53°, 127°, 53°, 127°
Perpendicular and parallel
PERPENDICULAR lines cross at exactly 90°. All four angles are right angles. PARALLEL lines never meet, however far you extend them — so they never cross at all. A line that crosses a pair of parallel lines is called a TRANSVERSAL.
Worked examples
One angle at a crossing is 53° — find the one beside it
- Angles on a straight line add to 180°.
- 180 − 53 = 127.
Answer: 127°
Find all four angles from one 53°
- Opposite angles are equal, so one more is 53°.
- The other two are each 180 − 53 = 127°.
Answer: 53°, 127°, 53°, 127°
What are opposite angles?
- Two lines crossing make four angles.
- The ones directly across from each other are opposite.
Answer: They are always equal
Perpendicular lines
- They cross at exactly 90°.
- So all four angles are right angles.
Answer: Four 90° angles
Do parallel lines cross?
- Parallel lines never meet, however far you extend them.
Answer: No — they never cross at all
One angle is 90° — what are the others?
- The angle beside it is 180 − 90 = 90°.
- Opposite angles match.
Answer: All four are 90°
Where you see this in real life
- A pair of scissors makes opposite angles: as the blades open, the angle at the tip and the angle at the handles stay equal.
- Road markings at a crossroads meet at 90° — perpendicular lines, four right angles.
Key words
- Intersect
- to cross
- Opposite angles
- angles across from each other where two lines cross; they are equal
- Perpendicular
- at right angles, 90°
- Parallel
- always the same distance apart and never meeting
- Transversal
- a line crossing a pair of parallel lines
- Straight line
- angles along one add up to 180°
- Right angle
- an angle of exactly 90°
- Vertex
- the point where lines meet