Question

Given: $\overline{\mathrm{AM}}\cong \overline{\mathrm{MB}};\overline{\mathrm{AD}}\cong \overline{\mathrm{BC}}$

$\angle \mathbf{MDC}\cong \angle \mathbf{MCD}$

Prove: $\overline{\mathrm{AC}}\simeq \overline{\mathrm{BD}}$

Short answer

$\mathrm{\Delta }\mathbf{ADM}\cong \mathrm{\Delta }\mathbf{BCM}$by SSS postulate then by corresponding parts of congruent triangles $\angle \mathbf{ADC}\cong \angle \mathbf{BCD}$.$\mathrm{\Delta }\mathbf{ADC}\cong \mathrm{\Delta }\mathbf{BCD}$by SAS postulate then corresponding parts of congruent triangles, $\overline{\mathrm{AC}}\cong \overline{\mathrm{BD}}$.

Step-by-step solution

Step 1. Description of step.

Consider$\mathrm{\Delta }ADM$ and $\mathrm{\Delta }BCM$. From the given figure it can be observed that $\overline{AD}\cong \overline{BC}$,$\overline{DM}\cong \overline{CM}$ and$\overline{AM}\cong \overline{MB}$ then by SSS postulate, $\mathrm{\Delta }ADM\cong \mathrm{\Delta }BCM$.

Step 2. Description of step.

As$\mathrm{\Delta }ADM\cong \mathrm{\Delta }BCM$ then by corresponding parts of congruent triangles$\angle ADM\cong \angle BCM$ and$\angle MDC\cong \angle MCD$ which implies that $\angle ADC\cong \angle BCD$.

Step 3. Description of step.

Consider$\mathrm{\Delta }ADC$ and $\mathrm{\Delta }BCD$. From the given figure it can be observed that $\overline{AD}\cong \overline{BC}$,$\angle ADC\cong \angle BCD$ and$\overline{CD}\cong \overline{CD}$ then by SAS postulate, $\mathrm{\Delta }ADC\cong \mathrm{\Delta }BCD$.

Step 4. Description of step.

As$\mathrm{\Delta }ADC\cong \mathrm{\Delta }BCD$ then by corresponding parts of congruent triangles, $\overline{AC}\cong \overline{BD}$.

Step 5. Write a paragraph proof.

$\mathrm{\Delta }ADM\cong \mathrm{\Delta }BCM$ by SSS postulate then by corresponding parts of congruent triangles $\angle ADC\cong \angle BCD$.$\mathrm{\Delta }ADC\cong \mathrm{\Delta }BCD$ by SAS postulate then corresponding parts of congruent triangles, $\overline{AC}\cong \overline{BD}$.