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A quadrilateral (four sided figure) has two sides which are parallel and equal in length - Leaving Cert Mathematics - Question 6B - 2013
Question 6B
A quadrilateral (four sided figure) has two sides which are parallel and equal in length. Prove that the quadrilateral is a parallelogram. In the parallelogram ABCD... show full transcript
Worked Solution & Example Answer:A quadrilateral (four sided figure) has two sides which are parallel and equal in length - Leaving Cert Mathematics - Question 6B - 2013
Step 1
Prove that the quadrilateral is a parallelogram.
Answer
Let the quadrilateral be WXYZ, where WX || ZY and |WX| = |ZY|.
To prove: WXYZ is a parallelogram.
- Join Z to Y and W to X.
- In triangles ZOY and DOW:
- We have |ZY| = |WX| (given)
- WX || ZY implies that ∠ZOW = ∠YOH (alternate interior angles)
- Thus, triangles ZOY and DOW are congruent by AAS criterion.
- Therefore, |ZO| = |OX| and |YO| = |OW|, meaning the diagonals intersect at their midpoints.
- Hence, WXYZ is a parallelogram.
Alternative proof:
- In triangle AMXZ, |ZX| = |YZ| (common to both).
- Thus, by the SAS congruence criteria, triangles AMXZ and AXZY are congruent.
- Since WX || ZY and |WX| = |ZY|, it follows that WZ || XY, confirming that WXYZ is a parallelogram.
Step 2
Prove that EBFD is a parallelogram.
Answer
In parallelogram ABCD:
- DE is perpendicular to AC, meaning angle ADE is 90°.
- BF is perpendicular to AC, meaning angle ABF is also 90°.
- Since both DE and BF are perpendicular to the same line AC, by the properties of parallel lines:
- DE || BF.
- Additionally, since EA || DB (since it is a parallelogram inherently), we can use the properties of transversals between parallel lines to establish that EBFD is a parallelogram.
Therefore, EBFD is proven to be a parallelogram.