Theorem

Parallel lines cut transversals proportionally

Geometry • Line • Parallel Lines and Transversals

Statement

If three or more parallel lines are cut by two transversals, the intercepts they make on one transversal are proportional to the corresponding intercepts on the other. In the triangle case, a line parallel to one side of a triangle divides the other two sides in the same ratio: if

D E ∥ B C

with

D

A B

and

E

A C

, then

\frac{A D}{D B} = \frac{A E}{E C}

. The converse holds in both forms.

How to use it

1The result is independent of the angles the transversals make, so no measurement of the figure is needed — only the ratios.
2Equal intercepts on one transversal force equal intercepts on the other; this is how a segment is divided into $n$ equal parts with ruler and compass alone.
3Use the converse to prove two lines parallel once the two ratios are known to be equal.
4Choose the pairing carefully: $\frac{A D}{D B} = \frac{A E}{E C}$ and $\frac{A D}{A B} = \frac{A E}{A C}$ are both true, but mixing a part with a whole is not.

Worked example

△ A B C

D

lies on

A B

and

E

A C

with

D E ∥ B C

. If

A D = 3

cm,

D B = 6

cm and

A E = 4

cm, find

E C

Step-by-step

1
$\frac{A D}{D B} = \frac{A E}{E C}$
Why: $D E ∥ B C$ , so the two sides are cut in the same ratio.
2
$\frac{3}{6} = \frac{4}{E C} \Rightarrow E C = 8$ cm

Answer

8

Watch out

Equal intercepts on one transversal require the parallels to be equally spaced — do not assume one transversal's intercepts carry over unchanged to the other, only their ratio does. The theorem says nothing about the lengths of the parallel segments themselves unless the transversals meet, as they do in a triangle.

Builds on

Also known as

basic proportionality theorem, BPT, Thales theorem, intercept theorem, equal intercept theorem, line parallel to one side

Parallel lines cut transversals proportionally

Statement

Worked example

Appears in past papers

2020 (II)

2020 (I)