This exam was adminstered in August 2023.

### August 2023 Geometry, Part IV

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

**35.** *In the diagram below of quadrilateral FACT, BR intersects diagonal AT at E,
AF || CT, and AF ≅ CT.
*

*Prove: (AB)(TE) = (AE)(TR)
*

**Answer: **

This question is a little different, because most proofs of this type usuablly rely on find two traingles to be congruent via SSS, SAS, etc., and then showing two parts to be congruent using CPCTC. This one doesn't do that. This one asks you to prove a multiplication statement is true, but these products are the results of cross-multiplying the numerators and denominators of a proportion. You need to show that the two triangles are similar, not congruent, and then cross multiply the sides because the product of the means will equal the product of the extremes.

Also, notice that you are only told that this is a "quadrilateral", not a parallelogram. This was the case in June, as well. I wonder if it's going to be a recurring theme going forward.

Your proof should look like this:

Statement | Reasons |

Quadrilateral FACT, BR intersects diagonal AT at E, AF || CT, and AF ≅ CT. | Given |

FACT is a parallelogram | A quadrilateral with one pair of sides that are parallel and congruent is a parallelogram |

∠ BAE ≅ ∠ RTE | Alternate Interior Angles |

∠ AEB ≅ ∠ TER | Vertical Angles |

△AEB ~ △RET | AA Postulate |

(AB) / (AE) = (TR) / (TE) | Corresponding sides of similar triangles are proportional |

(AB)(TE) = (AE)(TR) | The product of the means = the product of the extremes |

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.

Also note that I wrote "The product of the means = the product of the extremes" instead of "cross multiplication", which is in essence the same thing, except that it is an instruction and not a justification or Reason (theorem, postualte, etc.). You could argue this point all you want, but there is an example in the Model Response Set where the only point lost is for the final step where "Cross multiply" is written, and the Regents literally calls out this step as an incorrect reason.

**End of Exam**

How did you do?

Questions, comments and corrections welcome.

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