Question

Write the proofs in two-column form.

Given: $\angle \mathbf{1}\cong \angle \mathbf{2}\cong \angle \mathbf{3}$;$\overline{\mathrm{EN}}\cong \overline{\mathrm{DG}}$

Prove: $\angle \mathbf{4}\cong \angle \mathbf{5}$

Short answer

The proofs in two-column form is:

Statements	Reasons
$\angle 1\cong \angle 2\cong \angle 3$	Given
$\overline{ME}\cong \overline{MD}$	If two angles of a triangle are congruent, then sides opposite to those angles are congruent.
$\overline{EN}\cong \overline{DG}$	Given
$\angle MEN\cong \angle MDG$	As,$\angle 1\cong \angle 3$
$△MEN\cong △MDG$	By SAS postulate
$\angle 4\cong \angle 5$	By CPCT

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Write the theorem 4-2.

The theorem 4-2 states that if two angles of a triangle are congruent, then the sides opposite to those angles are congruent.

Step 3. Description of step.

It is given that$\angle 1\cong \angle 2\cong \angle 3$ and .

As, $\angle 1\cong \angle 2$, therefore by using the theorem 4-2, $\overline{ME}\cong \overline{MD}$.

As, $\angle 1\cong \angle 2\cong \angle 3$, therefore, $\angle 1\cong \angle 3$.

That implies, $\angle MEN\cong \angle MDG$.

Therefore, in the triangles$△MEN$ and $△MDG$, it can be noticed that $\overline{ME}\cong \overline{MD}$,$\angle MEN\cong \angle MDG$ and $\overline{EN}\cong \overline{DG}$,

Therefore, the triangles$△MEN$ and$△MDG$are congruent by using the SAS postulate.

Therefore, by corresponding parts of congruent triangles, it can be said that $\angle 4\cong \angle 5$.

Step 4. Write the proof in two-column proof.

The proofs in two-column form is:

Statements	Reasons
$\angle 1\cong \angle 2\cong \angle 3$	Given
$\overline{ME}\cong \overline{MD}$	If two angles of a triangle are congruent, then sides opposite to those angles are congruent.
$\overline{EN}\cong \overline{DG}$	Given
$\angle MEN\cong \angle MDG$	As,$\angle 1\cong \angle 3$
$△MEN\cong △MDG$	By SAS postulate
$\angle 4\cong \angle 5$	By CPCT