Question

3 select a reason
start overline, A, F, end overline, \cong, start overline, C, E, end overline
AF
≅
CE


Transitive Property
4 select a reason
angle, G, C, E, \cong, angle, H, A, F∠GCE≅∠HAF

Transitive Property
5 select a reason
triangle, G, C, E, \cong, triangle, H, A, F△GCE≅△HAF

ASA
6 select a reason
start overline, G, C, end overline, \cong, start overline, H, A, end overline
GC
≅
HA


Transitive Property
7 select a reason
start overline, B, C, end overline, \cong, start overline, A, D, end overline
BC
≅
AD


Transitive Property
8 select a reason
9 select a reason
start overline, A, B, end overline, \parallel, start overline, C, D, end overline
AB
∥
CD


If two lines cut by a transversal form congruent corresponding angles, then the two lines are parallel
10 select a reason
angle, B, A, E, \cong, angle, D, C, F∠BAE≅∠DCF

Answer 1

I think some of the reasons given should be CPCTC (corresponding parts of congruent triangles) rather than Transitive. Using step 5 (△GCE ≅ △HAF by ASA), any corresponding sides or angles of those triangles follow by CPCTC. Here are the corrected reasons:

- 3. AF ≅ CE — Reason: CPCTC (AF and CE are corresponding sides of congruent triangles △GCE and △HAF)
- 4. ∠GCE ≅ ∠HAF — Reason: CPCTC (corresponding angles of the congruent triangles)
- 5. △GCE ≅ △HAF — Reason: ASA (given in your list)
- 6. GC ≅ HA — Reason: CPCTC (corresponding sides of the congruent triangles)
- 7. BC ≅ AD — Reason: (depends on prior work) — if this comes from a triangle congruence, use CPCTC; if it follows from equalities combined earlier, use Transitive Property.
- 8. (blank) — if this is the statement ∠BAE ≅ ∠DCF used to prove the lines parallel, its reason should be CPCTC (if those angles correspond in congruent triangles) or whatever prior result gives that angle equality.
- 9. AB ∥ CD — Reason: If two lines cut by a transversal form congruent corresponding angles, then the lines are parallel (converse of corresponding-angles result)
- 10. ∠BAE ≅ ∠DCF — Reason: If two parallel lines are cut by a transversal, corresponding angles are congruent (Corresponding Angles Postulate)

If you paste the earlier steps or the full proof, I can place the exact reason for step 7 and fill step 8 precisely.