Geometry Problems of the Day (Geometry Regents, August 2024 Part IV)

This exam was adminstered in August 2024 .

August 2024 Geometry, Part IV

A correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

35. In quadrilateral HOPE below, EH ≅ OP, EP ≅ OH, EJ ≅ OG, and TG and YJ are perpendicular to diagonal EO at points G and J, respectively.

Prove that TG ≅ YJ.

Answer:

This is a long proof. Look at what you are given and what you need to find.

To show that TG = YJ, you need to show that triangle GET ≅ triangle JOY. To show that, you need to use one of the triangle postulates or theorems.

You are given that there are perpendicular lines, which create right angles, and we can show that there's a parallelogram so there's another angle. That means we could do with with either ASA or AAS.

Statement	Reason
1. EH ≅ OP, EP ≅ OH, EJ ≅ OG, and TG and YJ are perpendicular to diagonal EO at points G and J, respectively.	1. Given
2. HOPE is a parallelogram	2. A quadrilateral with two pairs of congruent sides is a parallelogram.
3. Angle JOY ≅ angle GOT	3. Alternate interior angles
4. Angle EGT and angle OJY are right angles.	4. Definition of perpendicular lines
5. GJ ≅ GJ	5. Reflexive Property
6. GE ≅ JO	6. Subtration Postulate
7. Triangle GET ≅ Triangle JOY	7. ASA Postulate
8. TG ≅ YJ.	8. CPCTC

End of Part Exam

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