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Rule

Transitive Property of Parallel Lines

Two lines that are parallel to the same line are also parallel to each other. In other words, if line

k

is parallel to line

m

and line

l

is also parallel to line

m,

then line

k

must also be parallel to line

l .

Based on the diagram, the following implication holds true.

{k ∥ m l ∥ m \Rightarrow k ∥ l

This concept is fundamental in geometry and is often used in proofs and geometric reasoning.

Why

This property is a consequence of the definition of parallel lines. Two lines in a plane are considered parallel if they never intersect, meaning they are always the same distance apart. Therefore, if two lines are parallel to a third line, each of the lines is a constant distance from the third line.

This means that the two lines are also a constant distance from each other (equal the sum or difference of the two mentioned distances to the third line).

Proof

Consider an auxiliary transversal crossing the lines and the angles formed by this intersection.

Since lines

k

and

m

are parallel, by the Corresponding Angles Theorem the corresponding angles formed by the transversal are congruent. Therefore,

∠ 1

and

∠ 2

are congruent. Similarly, since

l

and

m

are parallel,

∠ 3

and

∠ 2

are congruent as corresponding angles.

∠ 1 ∠ 3 ≅ ∠ 2 ≅ ∠ 2

By the Transitive Property of Congruence, since

∠ 1 ≅ ∠ 2

and

∠ 3 ≅ ∠ 2,

then

∠ 1 ≅ ∠ 3 .

∠ 1 ≅ ∠ 2 ∠ 3 ≅ ∠ 2 \Rightarrow ∠ 1 ≅ ∠ 3

The congruence of angles formed by a transversal intersecting lines

k

and

l

implies that the lines are parallel. This follows from the Converse Corresponding Angles Theorem.

$k ∥ l$

This shows why the Transitive Property of Parallel Lines holds true.

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