Question

Write the name or statement of the definition, postulate, or theorem that justifies the statement about the diagram.

If $\angle 4\cong \angle 3$, then$\overleftrightarrow{DG}$ bisects $\angle BDE$.

Short answer

The name of the theorem which justifies the statement about the given diagram is angle bisector theorem.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Definition of angle bisector theorem.

The angle bisector theorem states that if $\overleftrightarrow{BX}$is the bisector of $\angle ABC$, then $\angle ABX=\frac{1}{2}\angle ABC$and $\angle XBC=\frac{1}{2}\angle ABC$.

Step 3. Write the statement of the definition that justifies the statement about the given diagram.

Given that $\angle 4\cong \angle 3$.

That implies it is given that the angles∠4 and∠3 are the congruent angles.

Therefore, the angles ∠4 and∠3 will have equal measures.

Therefore, $\angle 4=\angle 3$.

It can be noticed that:

$$\begin{gathered} \angle 4=\angle 3 \\ \angle 4+\angle 3=\angle 3+\angle 3 \\ \angle GDE+\angle BDG=2\angle 3 \\ \angle BDE=2\angle BDG \\ \frac{1}{2}\angle BDE=\angle BDG \end{gathered}$$

As $\angle BDG=\frac{1}{2}\angle BDE$, therefore by using the angle bisector theorem it can be said that$\overleftrightarrow{DG}$ bisects the angle∠BDE.

Therefore, the name of the theorem which justifies the statement about the given diagram is angle bisector theorem.