Question

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, the statements and the first reason are filled in for you. Use the dropdown menu in the Reason column to select the reason for the next statement. Once you have selected a reason, a new statement will appear in a new row below. This will continue until you have selected the reason for the final statement, and then the answer area will appear with a submit button. The diagram associated with this problem is located at the bottom of the page, below the proof, 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 select a reason
start overline, E, F, end overline, \cong, start overline, E, F, end overline
EF
≅
EF

Answer 1

Fill in the missing reasons as follows:

1. Given (already)

2. EF ≅ EF — Reason: Reflexive property (any segment is congruent to itself)

3. AB ≅ CD — Reason: Opposite sides of a parallelogram are congruent

4. AG ≅ CH — Reason: If congruent segments are subtracted from congruent segments, the remainders are congruent (subtract BG ≅ DH from AB ≅ CD)

5. ∠GAE ≅ ∠HCF — Reason: Opposite angles of a parallelogram are congruent (or angles at A and C are congruent)

6. ΔAGE ≅ ΔCHF — Reason: SAS (AG ≅ CH, AE ≅ FC, included angles ∠GAE ≅ ∠HCF)

7. GE ≅ HF — Reason: Corresponding parts of congruent triangles are congruent (CPCTC)

This yields GE ≅ HF as required.