June 2023 Geometry Regents Part IV

This exam was adminstered in January 2023. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

June 2023 Geometry, Part IV

A correct answer is worth 6 credits. Partial credit can be given for correct statements in the proof.

35. Given: Quadrilateral ABCD, AB ≅ CD, AB || CD, diagonal AC intersects EF at G, and DE ≅ BF

Prove: G is the midpoint of EF

Answer:

I basically called this question "evil". Why? Because of several things that were put in just to trip you up. Okay, so I was being a bit overdramatic, but there are quite a few things that need to be stated. You should have at least 10 steps in your proof. (I could have combined two of my statements and used the reason "Opposite sides in a parallelogram are both parallel and congruent.")

For starters, you are given that ABCD is a "quadrilateral". You are not given that it is a parallelogram. So before you do anything else, you're going to need to show that it is, because you will need those properties later.

To prove that G is the midpoint of EF, you need to show that EG ≅ GF. This means that you need to show that two triangles are congruent so that their corresponding parts will be (CPCTC).

The two traingles that must be congruent are AGE and CGF. If we first prove that ABCD is a parallelogram, we can show that AE ≅ FC, and use alternate interior angles and vertical angles to set up AAS.

Your proof should look like this:

Statement	Reasons
AB ≅ CD, AB || CD, diagonal AC intersects EF at G, and DE ≅ BF	Given
ABCD is a parallelogram	A quadrilateral with one pair of sides that are parallel and congruent is a parallelogram
AD ≅ BC	The opposite sides of a parallelogram are congruent.
AE ≅ CF	Subtraction Postulate
Angle AGE ≅ angle CGF	Vertical Angles
AD || CB	Opposite sides of a parallelogram are parallel
Angle EAG ≅ angle FCG	Alternate interior angles
Triangle AGE ≅ triangle CGF	AAS
EG ≅ FG	CPCTC
G is the midpoint of EF	If a point divides a segment into two congruent parts, it must be the midpoint.

Each statement is important. If you leave any out, you will lose one credit. However, if you have a couple of statements correct and the semblance of a proof, you should still earn two points for the question.

End of Exam

How did you do?

Questions, comments and corrections welcome.

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