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 9B(a) - 2010
Question 9B(a)
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 9B(a) - 2010
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
Use the fact that B'F' = |BC| = |AB|.
By assumption, |AB| = |BC|.
Since AE || DE and FB || AB, we have alternate angles: |∠ABE| = |∠E'FB'|.
By vertically opposite angles, |∠A'E'B| = |∠F'E'B|.
Therefore, triangles AMEB' and FB'E' are congruent by ASA (Angle-Side-Angle).
Thus, |AE'| = |F'E'|.
Also, by the property of parallelograms, |AE'| = |DE| and |F'E'| = |FE|.
Thus, we conclude that |DE| = |EF|.
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