Geometry Problems of the Day (Geometry Regents, January 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, January 2012

Part IV: Each correct answer will receive 6 credits. Partial credit is available.


38. In the diagram below of quadrilateral ABCD,AD ≅ BC and ∠DAE ≅ ∠BCE.

Line segments AC, DB, and FG intersect at E.

Prove: Triangle AEF ≅ triangle CEG

Answer:

There is enough here to show that ABCD is a parallelogram, if that becomes important. There are properties that could be used, the most important of which is probably that the diagonals will bisect each other.

You are given a pair of congruent angles. You have a pair of vertical angles. You can show that the contained sides are parallel. So you can prove congruency through ASA.

This is not an obvious one. But then again, it's a Part IV problem.

Statement	Reason
1. AD ≅ BC, ∠DAE ≅ ∠BCE	Given
2. AD || BC	Alternate interior angles are congruent
3. ABCD is a parallelogram	A quadrilateral with one pair of sides that are both parallel and congruent is a parallelogram
4. AE ≅ CE	The diagonals of a parallelogram bisect each other
5. ∠AEF ≅ &∠CEG	Vertical angles are congruent
5. Triangle AEF ≅ triangle CEG	ASA

End of Exam.

More to come. Comments and questions welcome.

More Regents problems.

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