Question

Fill in the missing statement and reason of the proof below.

Given: A, B, C, DABCD is a parallelogram, start overline, B, G, end overline, \cong, start overline, D, H, end overline
BG
≅
DH
and start overline, A, E, end overline, \cong, start overline, F, C, end overline, .
AE
≅
FC
.

Prove: start overline, G, E, end overline, \cong, start overline, H, F, end overline
GE
≅
HF
.
In the proof shown in the table below, one of the steps in the middle is missing. Before completing the missing step, make sure to read the steps that come after it. To complete the missing step, first use the dropdown menu in the statement column to select the format of the statement for that step. Once you select a format, a statement will appear with input boxes and/or dropdowns to complete, and a dropdown menu will appear in the reason column. Once completed, press the submit button in the answer area at the bottom of the page. The diagram associated with this problem is located below the proof table, and is available as a tactile printout. There may be a clarifying note below the diagram.
Step Statement Reason
1
A, B, C, DABCD is a parallelogram
start overline, B, G, end overline, \cong, start overline, D, H, end overline
BG
≅
DH

start overline, A, E, end overline, \cong, start overline, F, C, end overline
AE
≅
FC

Given
2
start overline, E, F, end overline, \cong, start overline, E, F, end overline
EF
≅
EF

Reflexive Property
3
start overline, A, F, end overline, \cong, start overline, C, E, end overline
AF
≅
CE

Congruent segments added to congruent segments form congruent segments
4
start overline, A, D, end overline, \cong, start overline, B, C, end overline
AD
≅
BC

Opposite sides of a parallelogram are congruent
5
start overline, A, H, end overline, \cong, start overline, C, G, end overline
AH
≅
CG

Congruent segments subtracted from congruent segments form congruent segments
6
start overline, A, D, end overline, \parallel, start overline, B, C, end overline
AD
∥
BC

Opposite sides of a parallelogram are parallel
7
8
triangle, C, G, E, \cong, triangle, A, H, F△CGE≅△AHF
SAS
9
start overline, G, E, end overline, \cong, start overline, F, H, end overline
GE
≅
FH

Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Answer 1

Step 7 statement: ∠GCE ≅ ∠HAF

Reason: AD ∥ BC (step 6) and AC is a transversal, so corresponding (alternate interior) angles are congruent.