Question

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

Given: triangle, D, E, C, \cong, triangle, B, F, C, comma△DEC≅△BFC, angle, A, \cong, angle, C, comma∠A≅∠C, start overline, A, B, end overline, is perpendicular to, start overline, B, F, end overline
AB
⊥
BF
and start overline, A, D, end overline, is perpendicular to, start overline, D, E, end overline, .
AD
⊥
DE
.

Prove: A, B, C, DABCD is a rhombus.
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
triangle, D, E, C, \cong, triangle, B, F, C△DEC≅△BFC
angle, A, \cong, angle, C∠A≅∠C
start overline, A, B, end overline, is perpendicular to, start overline, B, F, end overline
AB
⊥
BF

start overline, A, D, end overline, is perpendicular to, start overline, D, E, end overline
AD
⊥
DE

Given
2
angle, A, D, E∠ADE is a right angle
Perpendicular lines form right angles
3
angle, A, B, F∠ABF is a right angle
Perpendicular lines form right angles
4 type of statement
5
angle, C, D, E, \cong, angle, C, B, F∠CDE≅∠CBF
Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
6
angle, A, D, C, \cong, angle, A, B, C∠ADC≅∠ABC
Congruent angles added to congruent angles form congruent angles
7
A, B, C, DABCD is a parallelogram
A, B, C, DABCD is a parallelogram because it is a quadrilateral with both pairs of opposite angles congruent
8
start overline, D, C, end overline, \cong, start overline, B, C, end overline
DC
≅
BC

Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
9
A, B, C, DABCD is a rhombus
A, B, C, DABCD is a rhombus because it is a parallelogram with one pair of consecutive sides congruent

Answer 1

Step 4 statement: ∠ADE ≅ ∠ABF

Reason: All right angles are congruent.