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CDS Mathematics · Lines, Angles and Polygons

Lines, Angles and Parallels

Angles on a straight line add to 180°, vertically opposite angles are equal, and a transversal makes equal corresponding and alternate angles with parallel lines.

All Lines, Angles and Polygons notesCDS Mathematics strategy

Why this matters

Twelve PYQs, none HARD, several read from a figure. Three facts settle almost all of them: a straight line is 180°, crossing lines make equal opposite angles, and parallels cut every transversal at the same angle — and in the same ratio when there are three of them.

Concept 1 of 2: Angles where lines meet

Two crossing lines make two pairs of equal angles, and any two neighbours make a straight line. So one angle fixes all four.

Definition

  • Adjacent angles on a straight line add to 180∘180^\circ; complementary angles add to 90∘90^\circ.
  • Vertically opposite angles are equal.
  • Points equidistant from two intersecting lines lie on the two angle bisectors: a pair of perpendicular lines.
  • nn lines, no two parallel and no three through one point, meet in (n2)\binom n2 points.

Intersection points

(n2)=n(n−1)2\binom{n}{2} = \dfrac{n(n - 1)}{2}

Worked example

Two lines cross so that one angle is four times its neighbour. Find all four angles.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (II) 2018 — Elementary Mathematics · Q87Easy

Example 1 · Lines, Angles and Polygons · Lines, Angles and Parallels

If two lines AB and CD intersect at O such that ∠AOC=5∠AOD\angle AOC = 5 \angle AOD, then the four angles at O are

Two bisectors, not one

Two intersecting lines make two angles, and each has a bisector. Points equidistant from the lines fill BOTH bisectors, so the locus is a pair of lines.
Drill 5 more on angles where lines meet

Concept 2 of 2: Parallels and transversals

A transversal crosses parallel lines at the same slant, so the angles it makes repeat at each crossing. Three parallel lines also slice any two transversals into pieces in the same ratio.

Definition

  • Corresponding and alternate angles are equal; co-interior angles add to 180∘180^\circ.
  • Lines parallel to the same line are parallel; in a plane, lines perpendicular to the same line are parallel.
  • Equal acute angles with a third line do NOT make two lines parallel: they can slope opposite ways.
  • Three parallels cut transversals in the same ratio: ABBC=DEEF\dfrac{AB}{BC} = \dfrac{DE}{EF}.

Intercept ratio

ABBC=DEEF\dfrac{AB}{BC} = \dfrac{DE}{EF}

Worked example

Three parallel lines cut one transversal into 44 cm and 66 cm, and another into xx and 99 cm. Find xx.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q27Moderate

Example 2 · Lines, Angles and Polygons · Lines, Angles and Parallels

Three parallel lines xx, yy and zz are cut by two transversals mm and nn. Transversal mm cuts the lines xx, yy, zz at PP, QQ, RR respectively; and Transversal nn cuts the lines xx, yy, zz at LL, MM, NN respectively. If PQ=3PQ = 3 cm, QR=9QR = 9 cm and MN=10.5MN = 10.5 cm, then what is the length of LMLM ?

Equal angles, opposite slopes

Two lines making 30∘30^\circ with a third can slope up and down; they are not parallel. Parallel needs equal corresponding angles on the same side.
Drill 5 more on parallels and transversals

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Watch out for (2)

Test yourself on Lines, Angles and Polygons

10 past CDS questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Lines, Angles and Polygons chapter testDrill all 12 questions