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Rule

Perpendicular Transversal Theorem

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line as well.

Based on the diagram, the following statement holds true.

If $ℓ_{1} ∥ ℓ_{2}$ and $t ⊥ ℓ_{1},$ then $t ⊥ ℓ_{2} .$

Proof

It is given that $ℓ_{1}$ and $ℓ_{2}$ are parallel lines and the transversal $t$ is perpendicular to the line $ℓ_{1} .$ This means that the lines $ℓ_{1}$ and $t$ intersect at a right angle.

Angles $∠ 1$ and $∠ 2$ are corresponding angles. By the Corresponding Angles Theorem, $∠ 1$ and $∠ 2$ are congruent. This means that $∠ 2$ is also a right angle.

Therefore, $t$ is perpendicular to $ℓ_{2} .$ This proves the theorem.

If $ℓ_{1} ∥ ℓ_{2}$ and $t ⊥ ℓ_{1},$ then $t ⊥ ℓ_{2} .$

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Proof