Question

Copy everything shown and write a two-column proof.
Given: $m\angle DBA=45$; $m\angle DEB=45$.

Prove:$\angle DBC\cong \angle FEB$

Short answer

The two column proof is:

Statements	Reasons
$m\angle DBA=45$,$m\angle DEB=45$	Given
$\angle DBA+\angle DBC=180$	Definition of supplementary angles
$\angle DBC=135$	By solving the equation $\angle DBA+\angle DBC=180$.
$\angle DEB+\angle FEB=180$	Definition of supplementary angles
$\angle FEB=135$	By solving the equation$\angle DEB+\angle FEB=180$
$\angle DBC\cong \angle FEB$	As, $\angle DBC=135$,$\angle FEB=135$

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Description of step.

It is being given that$m\angle DBA=45$ and $m\angle DEB=45$.

From the diagram it can be noticed that the angles$\angle DBA$ and$\angle DBC$ forms the linear pair.

Therefore, the sum of the angles$\angle DBA$ and$\angle DBC$ is $180$.

That implies, $\angle DBA+\angle DBC=180$.

Therefore, it can be obtained that:

$$\begin{gathered} \angle DBA+\angle DBC=180 \\ 45+\angle DBC=180 \\ \angle DBC=180-45 \\ \angle DBC=135 \end{gathered}$$

Therefore, the measure of the angle$\angle DBC$ is 135.

From the diagram, it can be noticed that the angles$\angle DEB$ and$\angle FEB$ forms the linear pair.

Therefore, the sum of the angles$\angle DEB$ and$\angle FEB$ is 180.

That implies, $\angle DEB+\angle FEB=180$.

Therefore, it can be obtained that:

$$\begin{gathered} \angle DEB+\angle FEB=180 \\ 45+\angle FEB=180 \\ \angle FEB=180-45 \\ \angle FEB=135 \end{gathered}$$

Therefore, the measure of the angle$\angle FEB$ is 135.

As, $\angle DBC=\angle FEB=135$, therefore $\angle DBC\cong \angle FEB$.

Hence proved.

Step 3. Write the two column-proof.

The two column proof is:

Statements	Reasons
$m\angle DBA=45$,$m\angle DEB=45$	Given
$\angle DBA+\angle DBC=180$	Definition of supplementary angles
$\angle DBC=135$	By solving the equation $\angle DBA+\angle DBC=180$.
$\angle DEB+\angle FEB=180$	Definition of supplementary angles
$\angle FEB=135$	By solving the equation$\angle DEB+\angle FEB=180$
$\angle DBC\cong \angle FEB$	As, $\angle DBC=135$,$\angle FEB=135$