Question

In the diagram, $\overrightarrow{BE}\perp \overleftrightarrow{AC}$and $\overrightarrow{BD}\perp \overrightarrow{BF}$. Find the measures of the following angles.

$m\angle CBF$	$m\angle EBF$	$m\angle DBE$	$m\angle DBA$	$m\angle DBC$
x	?	?	?	?

Short answer

The following table represents the required solution:

$m\angle CBF$	$m\angle EBF$	$m\angle DBE$	$m\angle DBA$	$m\angle DBC$
x	$90-x$	$x$	$90-x$	$90+x$

Step-by-step solution

Step 1. Observe the given information.

As the measurement of angle CBF is already given and the perpendicular relationship between the given lines represents that the measurement of $\angle ABE$,$\angle CBE$ and$\angle DBF$ is equal to 90°.

Step 2. Determine the value of m∠EBF, m∠DBE, m∠DBA and m∠DBC. 

According to the figure $\angle EBF$and $\angle CBF$makes the right $\angle EBC$so by using this determines the value of $\angle EBF$.

$$\begin{gathered} \angle EBC=\angle EBF+\angle CBF \\ 90=\angle EBF+x \\ \angle EBF=90-x \end{gathered}$$

According to the figure$\angle EBF$ and$\angle DBE$ makes the right$\angle DBF$so by using this determines the value of $\angle DBE$.

$$\begin{gathered} \angle DBF=\angle EBF+\angle DBE \\ 90=90-x+\angle DBE \\ \angle DBE=90-90+x \\ \angle DBE=x \end{gathered}$$

According to the figure$\angle DBA$ and$\angle DBE$ makes the right$\angle ABE$so by using this determines the value of $\angle DBA$.

$$\begin{gathered} \angle ABE=\angle DBA+\angle DBE \\ 90=\angle DBA+x \\ \angle DBA=90-x \end{gathered}$$

According to the figure the right$\angle DBF$ and$\angle CBF$ makes$\angle DBC$so by using this determines the value of $\angle DBC$.

$$\begin{gathered} \angle DBC=\angle DBF+\angle CBF \\ =90+x \end{gathered}$$

Step 3. Make the complete table.

Make the complete table by writing the value of obtained angles into their respective columns.

$m\angle CBF$	$m\angle EBF$	$m\angle DBE$	$m\angle DBA$	$m\angle DBC$
x	$90-x$	x	$90-x$	$90+x$