Geometry Problems of the Day (Geometry Regents, January 2013)

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, June 2013

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


38. The diagram below shows rectangle ABCD with points E and F on side AB. Segments CE and DF intersect at G, and ∠ADG ≅ ∠BCG.
Prove that AE ≅ BF

Answer:

To prove that AE ≅ BF, you need to show that AF is congruent to BE and then subtract EF from both of them. To do that, you need to show that triangles ADF and BCE are congruent. Since ABCD is a rectangle, you already know that angles A and B are right angles.

A better approach is to show that triangle HTM ≅ triangle HBA.

We know that this is an isosceles triangle, which gives us a pair of sides. We have medians which give us a second side. And angle H is congruent to itself with the reflexive property.

Statement	Reason
1. ABCD is a rectangle, ∠ADG ≅ ∠BCG	Given
2. Angle A ≅ angle B	All right angles are congruent
3. AD ≅ BC	Opposite sides in a rectangle are congruent
4. Triangle ADF ≅ triangle BCE	ASA
5. AF ≅ BE	CPCTC
5. EF ≅ EF	Reflexive Property
6. AE ≅ BF	Subtraction Property of Congruency

End of Exam.

More to come. Comments and questions welcome.

More Regents problems.

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