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Given: E is the midpoint of AB and CD little help please would be great

Given: E is the midpoint of AB and CD little help please would be great-example-1
User AllenJB
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1 Answer

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You need to prove that ΔAEC ≅ ΔBED

You know that point E is the midpoint of AB and CD

The midpoint divides a line segment into two congruent lines. This means that E divides AB into two congruent segments AE and EB, and it also divides CD into two congruent segments CE and ED, then we can conclude that:

1. AE=EB → Reason Midpoint Theorem

2. CE=ED → Reason Midpoint theorem

Both lines AB and CD intersect at point E, forming an X shape. The ingles formed inside the X are vertically opposite angles, this means that

3. ∠AEC= ∠BED → Reason Vertically opposite angles

Since E is the midpoint where both "diagonals of the parallelogram" AB and CD intersect, then sides AC and DB are parallel. If AB || CD, then ∠ACE and ∠BDE are alternate interiors angles as well as ∠CAE and ∠DBE. Alternate interior angles are congruent, so that:

4. ∠ACE=∠BDE → Reason alternate interior angles, AB || CD

5. ∠CAE=∠DBE → Reason alternate interior angles, AB || CD

Then → ΔAEC ≅ ΔBED by SAS and AAA

User Kaunteya
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