Question

Given: $\angle \mathbf{1}\cong \angle \mathbf{2}$;$\angle \mathbf{3}\cong \angle \mathbf{4};\angle \mathbf{5}\cong \angle \mathbf{6}$

Prove:$\overline{\mathrm{BC}}\cong \overline{\mathrm{ED}}$

Short answer

$\mathrm{\Delta }\mathbf{AGB}\cong \mathrm{\Delta }\mathbf{AFE}$by SAS postulate then by corresponding parts of congruent triangles $\overline{\mathrm{AB}}\cong \overline{\mathrm{AE}}$.$\mathrm{\Delta }\mathbf{ABC}\cong \mathrm{\Delta }\mathbf{AED}$by SAS postulate then corresponding parts of congruent triangles, $\overline{\mathrm{BC}}\cong \overline{\mathrm{ED}}$.

Step-by-step solution

Step 1. Description of step.

Consider$\mathrm{\Delta }AGB$ and $\mathrm{\Delta }AFE$. From the given information it can be observed that $\angle 1\cong \angle 2$,$\overline{AG}\cong \overline{AF}$ as they are the opposite sides to the angles$\angle 3\text{and}\angle 4$ also,$\angle 7\cong \angle 8$ then by SAS postulate, $\mathrm{\Delta }AGB\cong \mathrm{\Delta }AFE$.

Step 2. Description of step.

As$\mathrm{\Delta }AGB\cong \mathrm{\Delta }AFE$ then by corresponding parts of congruent triangles $\overline{AB}\cong \overline{AE}$.

Step 3. Description of step.

Consider $\mathrm{\Delta }ABC$and $\mathrm{\Delta }AED$. From the given figure it can be observed that $\overline{AC}\cong \overline{AD}$as they are the opposite sides to the angles $\angle 5\text{and}\angle 6$,$\angle 1\cong \angle 2$ and$\overline{AB}\cong \overline{AE}$ then by SAS postulate, $\mathrm{\Delta }ABC\cong \mathrm{\Delta }AED$.

Step 4. Description of step.

As$\mathrm{\Delta }ABC\cong \mathrm{\Delta }AED$ then by corresponding parts of congruent triangles, $\overline{BC}\cong \overline{ED}$.

Step 5. Write a paragraph proof.

$\mathrm{\Delta }AGB\cong \mathrm{\Delta }AFE$ by SAS postulate then by corresponding parts of congruent triangles $\overline{AB}\cong \overline{AE}$.$\mathrm{\Delta }ABC\cong \mathrm{\Delta }AED$ by SAS postulate then corresponding parts of congruent triangles, $\overline{BC}\cong \overline{ED}$.