Chapter 3.5, Problem 23WE

To determine

To calculate: the measure of $\angle K$ if Sides $\overline{AB}$ and $\overline{CD}$ are extended to meet at point $K$ .

Answer to Problem 23WE

The measure of $\angle K={108}^o$

Explanation of Solution

Given information:

  $ABCDEFGHIJ$ is a regular Decagon.

Sides $\overline{AB}$ and $\overline{CD}$ are extended to meet at point $K$ .

Calculation:

Use theorem $3-13,$ to find the sum of interior angles of regular Decagon $ABCDEFGHIJ$ as follows; consider this sum as $S$ :

  $\begin{array}{l}S=(n-2){180}^{\circ }\\ =(10-2){180}^{\circ }\\ =8\times {180}^{\circ }\\ ={1440}^{\circ }\end{array}$

Hence the measure of each interior angle of this Decagon is calculated as,

  $\frac{S}{n}=\frac{{1440}^{\circ }}{10}$

  $={144}^{\circ }$

If sides $\overline{AB}$ and $\overline{CD}$ are extended to meet at point $K$ then triangle $BKC$ will form and $\angle KBC,\angle KCB$ are exterior angles of triangle $BKC$ .

The measures of exterior angles of angles $\text{B}$ and $\text{C}$ can be calculated as;

  $\begin{array}{l}m\angle KBC={180}^{\circ }-m\angle B\\ ={180}^{\circ }-{144}^{\circ }\\ ={36}^{\circ }\end{array}$

Similarly,

  $\begin{array}{l}m\angle KCB={180}^{\circ }-m\angle C\\ ={180}^{\circ }-{144}^{\circ }\\ ={36}^{\circ }\end{array}$

Use angle sum property of triangles in triangle $KBC,$ and find measure of angle $K$ as follows:

  $\begin{array}{r}\hfill m\angle K+m\angle KBC+m\angle KCB={180}^{\circ }\\ \hfill m\angle K+36Kd+{36}^{\circ }={180}^{\circ }\\ \hfill m\angle K+{72}^{\circ }={180}^{\circ }\end{array}$

Subtract both sides by 72 degrees,

  $\begin{array}{l}m\angle K={180}^{\circ }-{72}^{\circ }\\ ={108}^{\circ }\\ m\angle K={108}^{\circ }\end{array}$

Hence, the measure of $\angle K={108}^o$