Question

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

1. ab ≅ bc ≅ cd ≅ da 7. cpctc
2. figure abcd is a rhombus 8. definition of a rhombus

o △abc ≅ △adc ≅ △bad ≅ △bcd
o △amc ≅ △cmb ≅ △cmd ≅ △amd
o △abc ≅ △adc ≅ △amd ≅ △bmc
o △amb ≅ △cmd ≅ △bad ≅ △bcd

Explanation:

Step1: Analyze the given information

We know that in parallelogram \(ABCD\), diagonals \(AC\) and \(BD\) bisect each other (\(\overline{AM}\cong\overline{MC}\), \(\overline{MB}\cong\overline{MD}\)) and the angles formed by the diagonals are right - angles (\(\angle AMB\cong\angle BMC\cong\angle CMD\cong\angle DMA\)).

Step2: Apply SAS congruency

For \(\triangle AMB\), \(\triangle BMC\), \(\triangle CMD\) and \(\triangle DMA\), we have two pairs of congruent sides (from the bisection of the diagonals) and the included right - angles are congruent. By the Side - Angle - Side (SAS) congruency theorem, we can conclude that these four triangles are congruent.

Answer:

\(\triangle AMB\cong\triangle CMB\cong\triangle CMD\cong\triangle AMD\)