Parallel Lines & Ratios in Triangles സമാന്തരവരകൾ
Chapter 6 leans on one result again and again: a line parallel to one side of a triangle cuts the other two sides in the same ratio. This page builds up to that result from the angles made by parallel lines, and also shows the reverse (converse) statement.
1. What are parallel lines?
Two straight lines in a plane are parallel (സമാന്തരം) if they never meet, however far you extend them. They keep the same distance apart everywhere and point in the same direction – they have the same slope. Railway tracks, the lines on a ruled page, opposite edges of a door – all parallel. We write AB ∥ CD and mark parallel lines with little arrows.
2. A transversal and the eight angles
A line crossing two parallel lines is called a transversal. It makes eight angles, and they come in named pairs. Click the buttons to see each pair, and move the slider to change the transversal.
- Corresponding angles (same position at both crossings, like ∠1 and ∠5) are equal.
- Alternate angles (inside the parallels, on opposite sides of the transversal, like ∠1 and ∠7) are equal.
- Co-interior angles (inside, on the same side, like ∠1 and ∠8) add up to 180°.
- Vertically opposite angles (like ∠1 and ∠3) are always equal – parallel or not.
3. The rules work backwards too
If a transversal makes equal alternate angles (or equal corresponding angles) with two lines, then those two lines must be parallel. This is how we prove lines are parallel, and it is used in Exercise 1, Question 3.
4. The key theorem: a parallel line divides the sides in the same ratio ത്രികോണഭാഗം
Take ΔABC and draw a line PQ parallel to BC, with P on AB and Q on AC. Then P divides AB in exactly the same ratio as Q divides AC:
APPB = AQQC (and also APAB = AQAC)Whatever the position, the two fractions are identical. Lengths in "cm" (40 px = 1 cm).
A ladder leaning on a wall with rungs parallel to the ground: each rung cuts the two rails in the same ratio. The shelves of a bookcase against a sloping attic wall, the horizontal rungs of a triangular roof frame, the equally spaced lines on a football pitch seen in perspective – all obey this rule.
5. The converse: same ratio ⇒ parallel
The statement also works in reverse. If a line cuts AB and AC in the same ratio, then it must be parallel to BC. Chapter 6 uses this to show that a certain line is parallel without measuring any angle.
Theorem: PQ ∥ BC ⇒ AP/PB = AQ/QC.
Converse: AP/PB = AQ/QC ⇒ PQ ∥ BC.
In the textbook this idea appears in the chapter സമാന്തരവരകൾ (Parallel lines) under ത്രികോണഭാഗം (dividing a triangle).
6. A construction: dividing a line into equal parts
Parallel lines let us divide any line segment into equal parts – or into any ratio – without measuring. Exercise 1, Question 3 of this chapter uses exactly this trick.
7. Check yourself
- With parallel lines: corresponding angles equal, alternate angles equal, co-interior angles add to 180°.
- Equal alternate (or corresponding) angles ⇒ the lines are parallel.
- A line parallel to one side of a triangle divides the other two sides in the same ratio; and a line dividing two sides in the same ratio is parallel to the third side.
- Parallel lines can divide a segment into equal parts or into any given ratio.