Chapter 2.5, Problem 5CE

To determine

Measures of the following given angles.

Answer to Problem 5CE

  $\angle EBF=90-x,\angle DBE=x,\angle DBA=90-x\text{and }\angle DBC=90+x$

Explanation of Solution

Given:

The given figures are:

  

  

Also, $\overleftrightarrow{BE}\perp \overleftrightarrow{AC}$ and $\overleftrightarrow{BD}\perp \overleftrightarrow{BF}$

As given, $\angle CBF=x$ and

  $\overleftrightarrow{BE}\perp \overleftrightarrow{AC}$ and $\overleftrightarrow{BD}\perp \overleftrightarrow{BF}$

So, $\angle ABE=\angle CBE=90\text{and}\angle DBF=90$

Now, from the figure

  $\begin{array}{l}\angle EBC=\angle EBF+\angle CBF\\ 90=\angle EBF+x\\ \Rightarrow \angle EBF=90-x\end{array}$

Further,

  $\begin{array}{l}\angle DBF=\angle EBF+\angle DBE\\ 90=90-x+\angle DBE\\ \angle DBE=90-90+x\\ \Rightarrow \angle DBE=x\end{array}$

Also,

  $\begin{array}{l}\angle ABE=\angle DBA+\angle DBE\\ 90=\angle DBA+x\\ \Rightarrow \angle DBA=90-x\end{array}$

Now,

  $\begin{array}{l}\angle DBC=\angle DBF+\angle CBF\\ =90+x\\ \Rightarrow \angle DBC=90+x\end{array}$

So,

  $\angle EBF=90-x,\angle DBE=x,\angle DBA=90-x\text{and }\angle DBC=90+x$