Prove that if three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal line - Leaving Cert Mathematics - Question 6A - 2015
Question 6A
Prove that if three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal line.
Diagram:
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Worked Solution & Example Answer:Prove that if three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal line - Leaving Cert Mathematics - Question 6A - 2015
Step 1
Given:
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Answer
AD || BE || CF , as in the diagram, with |AB| = |BC|.
Step 2
To prove:
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Answer
|DE| = |EF|.
Step 3
Construction:
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Answer
Draw AE' || DE, cutting EB at E' and CF at F'. Draw FB' || AB, cutting EB at B', as in the diagram.
Step 4
Proof:
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Answer
|B'F'| = |BC| (opposite sides in a parallelogram).
|B'F'| = |AB| (by assumption).
|∠BAE'| ≅ |∠EF'B'| (alternate angles).
|AE'B'| = |EF'B'| (vertically opposite angles).
∴ ∠ABE' is congruent to ∠FB'F'E'.
Therefore |AE'| = |F'E'|.
But |AE'| = |DE| and |F'E'| = |EF| (opposite sides in a parallelogram).
Thus, |DE| = |EF|.
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