Chapter 9, Problem 18CR

a.

To determine

To calculate: The measure of anglesD and E .

Answer to Problem 18CR

The measure of angle D is ${140}^{\circ }$ and measure of angle Eis ${160}^{\circ }$ .

Explanation of Solution

Given information:

  $\angle A={120}^{\circ }.$

  $\overrightarrow{BD}$ and $\overrightarrow{BE}$ trisect $\angle ABC$ .

  $\overrightarrow{CD}$ and $\overrightarrow{CE}$ trisect $\angle ACB$ .

Formula used:

The below property is used:

If two lines trisect angle, it will divide that angle into three equal parts.

Calculation:

  

  $\angle A={120}^{\circ }.$

Sum of angles in a triangle is ${180}^{\circ }$ .

  $\angle B={30}^{\circ }$ and $\angle C={30}^{\circ }$

  $\overrightarrow{BD}$ and $\overrightarrow{BE}$ trisect $\angle ABC$ .

  $\begin{array}{l}\angle ABD={10}^{\circ }\\ \angle DBE={10}^{\circ }\\ \angle EBC={10}^{\circ }\end{array}$

  $\overrightarrow{CD}$ and $\overrightarrow{CE}$ trisect $\angle ACB$ .

  $\begin{array}{l}\angle ACD={10}^{\circ }\\ \angle DCE={10}^{\circ }\\ \angle ECB={10}^{\circ }\end{array}$

In $\Delta BCD$ ,

  $\begin{array}{l}\angle B+\angle D+\angle C={180}^{\circ }\\ \therefore {20}^{\circ }+\angle D+{20}^{\circ }={180}^{\circ }\\ \therefore \angle D={180}^{\circ }-{40}^{\circ }\\ \therefore \angle D={140}^{\circ }\end{array}$

In $\Delta BCE$ ,

  $\begin{array}{l}\angle B+\angle E+\angle C={180}^{\circ }\\ \therefore {10}^{\circ }+\angle E+{10}^{\circ }={180}^{\circ }\\ \therefore \angle E={180}^{\circ }-{20}^{\circ }\\ \therefore \angle E={160}^{\circ }\end{array}$

b.

To determine

To find:The three angles are in arithmetic progression.

Answer to Problem 18CR

Yes, the three angles are in arithmetic progression

Explanation of Solution

Given information:

The measures of $\angle A,\angle B,$ and $\angle C$ are ${120}^{\circ },{140}^{\circ }$ and ${160}^{\circ }$ respectively.

An arithmetic progression is a sequence of terms in which the difference between any two consecutive terms is always the same. (For example, 1, 5, 9, 13 is an arithmetic progression because the difference between any two consecutive terms is 4.)

  

The measures of $\angle A,\angle B,$ and $\angle C$ are ${120}^{\circ },{140}^{\circ }$ and ${160}^{\circ }$ respectively.

  ${120}^{\circ },{140}^{\circ }$ and ${160}^{\circ }$ is an arithmetic progression because the difference between any two consecutive terms is ${20}^{\circ }$ .