Question

Name the definition or state the theorem that justifies the statement about the diagram.

If $\angle ABE\cong \angle EBC$, then $\overleftrightarrow{AC}\perp \overrightarrow{BE}$.

Short answer

The used definition is the definition of perpendicular lines.

Step-by-step solution

Step 1. Observe the given statement and figure.

The provided relation between the lines is the result of the given relation between the angles.

Step 2. Identify the definition or theorem.

As the provided angles are congruent and the given figure represents that the line ABC represents itself as a straight line so the measurement of both the given angles will be equal to 180° so make an equation by using this.

$\angle ABE+\angle EBC=180°$

Substitute$\angle ABE$ in place of $\angle ABE$into the equation for this the given relation between these angles can be used. After it determines the value of $\angle ABE$.

$$\begin{gathered} \angle ABE+\angle ABE=180° \\ 2\angle ABE=180° \\ \angle ABE=90° \end{gathered}$$

So, both the angles are the right angles which means that $\overleftrightarrow{AC}\perp \overleftrightarrow{BE}$.

The observation of the statement returns the definition of perpendicular lines that is the relation between two lines is perpendicular when the measurement of angles formed by those lines is equal to each other.

Step 3. State whether the definition is true or not.

As the definition of the perpendicular line matches with the provided statement so the definition of perpendicular lines is the required definition.