Question

Given: A, B, C, DABCD is a rhombus.

Prove: triangle, A, E, B△AEB is a right triangle.
In the table below, the first step of the proof is completed for you. To complete each following step, first use the dropdown menu in the "Statement" column to select the format of a statement. 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 you complete these inputs and dropdowns, navigate back to the first column and press the "try" button. If your statement and reason are valid, a new row will appear below. This process will continue until you have filled in a statement and reason for what you were asked to prove, and then the "try" button will cause the answer area to appear with a submit button. 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 rhombus
Given
2
start overline, A, B, end overline, \cong, start overline, B, C, end overline
AB
≅
BC

All sides of a rhombus are congruent
3
start overline, A, D, end overline, \cong, start overline, D, C, end overline
AD
≅
DC

All sides of a rhombus are congruent. What are the remaining steps, statements, and reasons for this question.

Answer 1

Continue the proof with these statements and reasons:

4. AB ≅ AD
Reason: All sides of a rhombus are congruent.

5. AB ∥ CD
Reason: A rhombus is a parallelogram; opposite sides of a parallelogram are parallel.

6. BC ∥ AD
Reason: A rhombus is a parallelogram; opposite sides of a parallelogram are parallel.

7. AE ≅ CE and BE ≅ DE
Reason: Diagonals of a parallelogram bisect each other.

8. △ABE ≅ △CBE
Reason: SSS (AB ≅ CB, AE ≅ CE, BE ≅ BE).

9. ∠ABE ≅ ∠EBC
Reason: CPCTC (corresponding parts of congruent triangles).

10. △ABE ≅ △ADE
Reason: SSS (AB ≅ AD, BE ≅ DE, AE ≅ AE).

11. ∠BAE ≅ ∠EAD
Reason: CPCTC (so AC bisects ∠A).

12. In △ABE, ∠AEB = 180° − (∠BAE + ∠ABE)
Reason: Triangle angle sum theorem.

13. ∠BAE = 1/2 ∠A and ∠ABE = 1/2 ∠B
Reason: From steps 11 and 9 (each diagonal bisects the corresponding vertex angle).

14. ∠A + ∠B = 180°
Reason: Consecutive angles in a parallelogram are supplementary.

15. Therefore ∠AEB = 180° − 1/2(∠A + ∠B) = 180° − 1/2(180°) = 90°.
Reason: Substitution from steps 12–14.

Conclusion: △AEB is a right triangle (∠AEB = 90°).