Chapter 2.5, Problem 4CE

To determine

Measures of the following given angles.

Answer to Problem 4CE

  $\angle EBF=50,\angle DBE=40,\angle DBA=50\text{and }\angle DBC=130$

Explanation of Solution

Given:

The given figures are:

  

  

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

As given, $\angle CBF=40$ 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

  $\begin{array}{l}\angle EBC=\angle EBF+\angle CBF\\ 90=\angle EBF+40\\ \angle EBF=90-40\\ \Rightarrow \angle EBF=50\end{array}$

Further,

  $\begin{array}{l}\angle DBF=\angle EBF+\angle DBE\\ 90=50+\angle DBE\\ \angle DBE=90-50\\ \Rightarrow \angle DBE=40\end{array}$

Also,

  $\begin{array}{l}\angle ABE=\angle DBA+\angle DBE\\ 90=\angle DBA+40\\ \angle DBA=90-40\\ \Rightarrow \angle DBA=50\end{array}$

Now,

  $\begin{array}{l}\angle DBC=\angle DBF+\angle CBF\\ =90+40\\ \Rightarrow \angle DBC=130\end{array}$

So,

  $\angle EBF=50,\angle DBE=40,\angle DBA=50\text{and }\angle DBC=130$