geometry help please

Question

. Given: Quadrilateral EFHJ is a parallelogram, <1 is congruent to <2

Prove: Segment KH is congruent to segment EG

Answer 1

Hi Antonia! I haven't done "Geometry style" proofs in a while, but I can try to outline the ideas that would go into demonstrating this. I apologize in advance for not remembering some of the theorems, but hopefully this can help you get moving in the right direction.

1. Right from the get-go, the problem tells us that < KHJ ≅ < GEF.
2. By definition of a parallelogram, we know that < KJH ≅ < EFG.
3. By that same definition, we know that the line segments JH and EF are of equal length, and are thus congruent.
4. This gives us two angles and their included side for both triangle Δ HJK and for triangle Δ EFG, which is enough to prove that the two triangles are congruent (by angle-side-angle, or ASA congruence).
5. Because triangles Δ HJK and Δ EFG are congruent, each of their corresponding sides and each of their corresponding angles must be congruent.
6. Finally, because the line segments KH and EG are corresponding sides of the aforementioned triangles, they must be congruent (by definition of congruent triangles). QED.

Again, I apologize for not formatting this more as a "proofs class" proof rather than as a "geometry class" proof. I am a bit out of practice with the geometry style, but I wanted to try to help get you going on it. (If anyone else out there has a better version, I would love to see it!) If you have any questions, let me know and I can try to clarify.

Hope this helps!

Tim W