A quadrilateral whose diagonals bisect each other is a parallelogram.
Step-by-step solution
To find: Prove that ABCD is a parallelogram
Idea: The original proof used ∆AED ≅ ∆CEB to show the diagonals bisect each other. Run it backwards: now the halves of the diagonals are equal, so the same triangles are congruent by SAS, and their equal angles give parallel sides.
- In ∆AED and ∆CEB: EA = EC and ED = EB (given), ∠AED = ∠CEB (vertically opposite angles). So ∆AED ≅ ∆CEB (SAS).1 mark
- Hence ∠DAE = ∠BCE (CPCT). These are alternate angles made by transversal AC with lines AD and BC, so AD ‖ BC.1 mark
- In the same way, ∆EAB ≅ ∆ECD (SAS, with ∠AEB = ∠CED), so ∠BAE = ∠DCE and AB ‖ DC. Both pairs of opposite sides are parallel: ABCD is a parallelogram.1 mark
Answer to write in the exam
In ∆AED and ∆CEB: EA = EC, ED = EB (given), ∠AED = ∠CEB (vertically opposite angles)
∴ ∆AED ≅ ∆CEB (SAS) ⇒ ∠DAE = ∠BCE (CPCT) ⇒ AD ‖ BC (alternate angles equal)
Similarly ∆EAB ≅ ∆ECD (SAS) ⇒ ∠BAE = ∠DCE ⇒ AB ‖ DC
∴ ABCD is a parallelogram.
Common mistakes that cost marks
- Forgetting to state the included angle: SAS needs ∠AED = ∠CEB (vertically opposite).
- Taking ∠DAE and ∠BCE as “corresponding” angles. They are alternate angles on transversal AC.
- Assuming the diagonals are equal. They only bisect each other; equal diagonals would make a rectangle.
How this can come in the exam
Two straight sticks of different lengths are tied together at their midpoints and the four ends are joined by string. Show that the string forms a parallelogram.
Show answer
The sticks are the diagonals of the four-sided figure and, being tied at their midpoints, they bisect each other (1 mark). A quadrilateral whose diagonals bisect each other is a parallelogram (1 mark).Try one yourself
The diagonals of quadrilateral WXYZ meet at O, with OW = OY = 5 cm and OX = OZ = 3 cm. If ∠XWY = 32°, find ∠WYZ.
Show answer
The diagonals bisect each other, so WXYZ is a parallelogram and WX ‖ ZY. ∠WYZ and ∠XWY are alternate angles: ∠WYZ = 32°.
More questions like this
- A quadrilateral with one pair of equal and parallel opposite sides is a parallelogram.
- Suppose in a quadrilateral ABCD we have AB ‖ DC and AB = DC. Let E be the intersection point of the diagonals AC and BD. (Why must the diagonals intersect?)
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- The diagonal AC of a parallelogram ABCD bisects ∠A. Show that it also bisects ∠C and that ABCD is a rhombus.
- The following questions examine converses of true properties. Answer them with Yes or No. If your answer is No, what extra condition can you add so that the answer becomes Yes?
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