This exam was adminstered in August 2025.
August 2025 Geometry, Part IV
Each correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.
35. In quadrilateral ABCD below, side CD is extended through D to point E such that AFD and BFE bisect each other, and DE ≅ DC.
Prove ABCD is a parallelogram.
Answer:
The rule you need to remember here is that a quadrilateral with one pair of opposite sides that are both parallel and congruent is a parallelogram.
We can prove this by showing that the two triangles in the image are congruent. This will show that the AB || CD and then the transitive property will show that AB ≅ CD.
| Statement | Reason |
| 1. quadrilateral ABCD with side CD is extended through D to point E such that AFD and BFE bisect each other, and DE ≅ DC. | Given. |
| 2. AF ≅ FD and EF ≅ FB. | Definition of bisector. |
| 3. ∠AFB ≅ ∠DFE | Vertical angles are congruent. |
| 4. △DFE ≅ △AFB | SAS Postulate |
| 5. ∠DEF ≅ ∠ABF | CPCTC |
| 6. AB || EC | If two lines are cut by a transversal and the Alternate Interior Angles are congruent, then the lines are paralllel. |
| 7. AB ≅ DE | CPCTC |
| 8. AB ≅ DC | Transitive Property / Substitution |
| 9. ABCD is a parallelogram | In a quadrilateral, if two opposite sides are parallel and congruent then the quadrilateral in a parallelogram. |
One curious note:
In Step 8, I would've stated the reason as "the transitive property". It's curious that in the all the correct solutions in the model response set, the word "substitution" is used. In fact, there's only one example that uses the phrase "Transitive Property", but it uses it incorrectly -- there was a step missing before that. This has me questioning if that is a sufficient "Reason" for full credit. I like it better than "substitution" myself, which is also logical, but sounds more algebraic to me.
End of Part Exam
How did you do?
Questions, comments and corrections welcome.
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