Geometry • Topic 6

Parallel Lines

Parallel lines are lines in the same plane that never intersect. They are essential for translating angle information across a diagram.

Definition and Axiom

Definition (Parallel)

Two lines

ℓ

and

m

are parallel (

ℓ ∥ m

) if they lie in the same plane and have no common point.

:::axiom[Playfair's Axiom] Given a line $ℓ$ and a point $P$ not on $ℓ$ , there is exactly one line through $P$ parallel to $ℓ$ . :::

Transversals and Angles

A transversal is a line that intersects two or more coplanar lines.

[Image of Parallel lines cut by a transversal showing angle relationships]

If two parallel lines are cut by a transversal:

Alternate Interior Angles: Equal ( $α = β$ ). ("Z" shape) [cite_start][cite: 54]
Corresponding Angles: Equal. ("F" shape)
Consecutive Interior Angles: Supplementary (Sum to $18 0^{\circ}$ ). ("C" shape)

Proportionality (Thales' Theorem)

Theorem (Intercept Theorem)

If two secants

S_{1}, S_{2}

intersect three parallel lines at

A, B, C

and

A^{'}, B^{'}, C^{'}

respectively, then the intercepts are proportional:

\frac{A B}{BC} = \frac{A ^{'} B ^{'}}{B ^{'} C ^{'}}

This implies that a line parallel to one side of a triangle divides the other two sides proportionally.

Practice Problems

Exercise (Problem 1)

Prove that the sum of the angles in a triangle is

18 0^{\circ}

using parallel lines. (Hint: Draw a line through one vertex parallel to the opposite side).

Exercise (Problem 2)

In a trapezoid

A BC D

(

A B ∥ C D

), the bisectors of adjacent angles

∠ A

and

∠ D

intersect at a point

P

. Prove that

P

lies on the median (midline) of the trapezoid.

Exercise (Problem 3)

Three parallel lines

ℓ_{1}, ℓ_{2}, ℓ_{3}

are cut by two transversals. The segments on the first transversal have lengths 3 and 5. If the total length of the segment on the second transversal is 24, find the lengths of its parts.