Chapter 7, Problem 19RP

a

To determine

To calculate : The value of $m\angle ABF$

Answer to Problem 19RP

The value of $\angle ABF={32.5}^0$

Explanation of Solution

Given information : The following information has been given

  $\Delta ABC$ is isosceles

  $\overrightarrow{BE}$ bisects $\angle ABC$

  $\overrightarrow{CE}$ bisects $\angle FCD$

  $\angle A={50}^0$

Formula used : The angles of any triangle add up to ${180}^0$

Calculation : We know that $\Delta ABC$ is isosceles. Hence, if $\angle A={50}^0$ ,

  $\angle ACB\cong \angle ABC=\frac{1}{2}({180}^0-{50}^0)={65}^0$

Now, we know that $\overrightarrow{BE}$ bisects $\angle ABC$ , hence

  $\angle ABF\cong \angle FBC=\frac{1}{2}({65}^0)={32.5}^0$

Thus, the value of $\angle ABF={32.5}^0$

b

To determine

To calculate : The value of $m\angle BCE$ is to be calculated

Answer to Problem 19RP

The value of $\angle BCE={122.5}^0$

Explanation of Solution

Given information : The following information has been given

  $\Delta ABC$ is isosceles

  $\overrightarrow{BE}$ bisects $\angle ABC$

  $\overrightarrow{CE}$ bisects $\angle FCD$

  $\angle A={50}^0$

Formula used : The angles of any triangle add up to ${180}^0$

Calculation : We know that

  $\angle ABF\cong \angle FCE=\frac{1}{2}({180}^0-{65}^0)={57.5}^0$

Thus, we can say that

  $\angle BCE\cong \angle BCF+\angle FCE$

Hence,

  $\angle BCE={65}^0+{57.5}^0={122.5}^0$

c

To determine

To calculate : The value of $m\angle E$ is to be calculated

Answer to Problem 19RP

The value of $m\angle E={155}^0$

Explanation of Solution

Given information : The following information has been given

  $\Delta ABC$ is isosceles

  $\overrightarrow{BE}$ bisects $\angle ABC$

  $\overrightarrow{CE}$ bisects $\angle FCD$

  $\angle A={50}^0$

Formula used : The angles of any triangle add up to ${180}^0$

Calculation : We know that in triangle BEC,

  $m\angle E\cong {180}^0-(\angle EBC+\angle ECB)={180}^0-({32.5}^0+{122.5}^0)$

Thus, the value of $m\angle E={155}^0$