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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
