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Diagonals AC and BD form right angles at point M in parallelogram ABCD. Prove ABCD is a rhombus.


Parallelogram A B C D is shown. Diagonals are drawn from point A to point C and from point B to point D and intersect at point M. Sides A B and D C are parallel and sides B C and A D are parallel. A square is drawn around point M.



Statements


Reasons

1. ABCD is a parallelogram 1. given

2. ∠AMB, ∠BMC, ∠CMD, and ∠DMA are right angles 2. given

3. ∠AMB ≅ ∠BMC ≅ ∠CMD ≅ ∠DMA 3. right angles are congruent

4. AC bisects BD;

BD bisects AC; 4. diagonals of a parallelogram bisect each other

5. AM ≅ MC, MB ≅ MD 5. definition of a bisector

6. ? 6. SAS congruency theorem

7. AB ≅ BC ≅ CD ≅ DA 7. CPCTC

8. figure ABCD is a rhombus 8. definition of a rhombus

△ABC ≅ △ADC ≅ △BAD ≅ △BCD

△AMB ≅ △CMB ≅ △CMD ≅ △AMD

△ABC ≅ △ADC ≅ △AMD ≅ △BMC

△AMB ≅ △CMD ≅ △BAD ≅ △BCD