Chapter 2.5, Problem 8WE

To determine

To state: The theorem that justifies “If $\angle EBC\cong \angle ABE$ , then $AC\perp BE$ ” for the given diagram.

Answer to Problem 8WE

The theorem is:

If two line makes angle of ${90}^{\circ }$ at intersection then both are perpendicular to each other.

Explanation of Solution

Given information:

  $\angle EBC\cong \angle ABE$

Since $\overleftrightarrow{ABC}$ is a straight line.

Therefore, $\angle ABE+\angle EBC={180}^{\circ }----(1)$

Since $\angle ABE=\angle EBC----(2)$ .

Therefore, putting value of $\angle EBC$ in equation (1),

  $\begin{array}{l}\angle ABE+\angle ABE={180}^{\circ }\\ 2\angle ABE={180}^{\circ }\\ \angle ABE={90}^{\circ }\end{array}$

Which implies, $\angle EBC={90}^{\circ }$ .

Hence, $AC\perp BE$

The theorem is:

If two line makes angle of ${90}^{\circ }$ at intersection then both are perpendicular to each other.