E is the midpoint of DA, and F is the midpoint of DB. This means EF is a midsegment of the triangle. It is half as long as AB, and EF is also parallel to AB.
When we extend EF to form segment EC, it is now twice as long compared to EF. So that shows EC is the same length as AB, in other words, EC = AB. Also, EC is parallel to AB. This works due to points E, F, and C being on the same line.
Since EC || AB, we know that angle EBA = angle BEC. These are alternate interior angles.
From the reflective property, we can say EB = EB
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Using these three facts
- EC = AB
- angle EBA = angle BEC
- EB = EB
to prove that triangle EBA is congruent to triangle BEC. The next step is to use CPCTC (corresponding parts of congruent triangles are congruent) to lead to
angle BEA = angle EBC
we can use the alternate interior angle theorem converse to conclude that EA is parallel to BC.
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In summary, we have proven:
- EC is parallel to AB
- EA is parallel to BC
By the definition of what it means to be a parallelogram, we have proven quadrilateral EABC is a parallelogram. In short, we've shown that the opposite sides are parallel.
