Chapter 4.4, Problem 25WE

(a)

To determine

To Find: The measure of angle $\angle 7$ .

Answer to Problem 25WE

  $m\angle 7=90°$

Explanation of Solution

Given information:

In the given figure, $m\angle 1=23°$ .

  

Calculation:

Let’s name the vertices as A, B, C, D, E.

Since, in triangle ABC adjacent sides are equal therefore angle opposite to these sides are also equal, that is,

  $\begin{array}{l}m\angle 1=m\angle 2\\ m\angle 2=23°\end{array}$

Since, the sides DE and AB are parallel so interior angles are supplementary.

  $m\angle 3+m\angle 4=180°$

Sum of angles in triangle ABC ,

  $\begin{array}{c}m\angle 1+m\angle 2+m\angle 3=180°\\ 23°+23°+m\angle 3=180°\\ m\angle 3=134°\end{array}$

Substituting $m\angle 3=134°$ in the equation $m\angle 3+m\angle 4=180°$ gives,

  $\begin{array}{l}134°+m\angle 4=180°\\ m\angle 4=46°\end{array}$

Since, in triangle CDE adjacent sides are equal, therefore angle opposite to these sides are also equal, that is,

  $m\angle 5=m\angle 6$

Also, sum of angles in triangle is CDE is equal to $180°$ .

  $\begin{array}{l}m\angle 4+m\angle 5+m\angle 6=180°\\ 46°+m\angle 5+m\angle 5=180°\\ 2m\angle 5=134°\\ m\angle 5=67°\end{array}$

So, $m\angle 6=m\angle 5=67°$ .

Line AD is a straight line so,

  $\begin{array}{l}m\angle 2+m\angle 5+m\angle 7=180°\\ 23°+67°+m\angle 7=180°\\ m\angle 7=90°\end{array}$

Therefore, the measure of $\angle 7$ is $90°$ .

(b)

To determine

To Find: The measure of angle $\angle 7$ .

Answer to Problem 25WE

  $m\angle 7=90°$

Explanation of Solution

Given information:

In the given figure, $m\angle 1=k$ .

  

Calculation;

Let’s name the vertices as A, B, C, D, E.

Since, in triangle ABC adjacent sides are equal therefore angle opposite to these sides are also equal, that is,

  $\begin{array}{l}m\angle 1=m\angle 2\\ m\angle 2=k\end{array}$

Since, the sides DE and AB are parallel, so interior angles are supplementary.

  $m\angle 3+m\angle 4=180°$

Sum of angles in triangle ABC is $180°$ .

  $\begin{array}{l}m\angle 1+m\angle 2+m\angle 3=180°\\ k+k+m\angle 3=180°\\ m\angle 3=180°-2k\end{array}$

Substitute $\angle 3=180°-2k$ in the equation $m\angle 3+m\angle 4=180°$ .

  $\begin{array}{l}180°-2k+m\angle 4=180°\\ m\angle 4=2k\end{array}$

Since, in triangle CDE adjacent sides are equal, therefore angle opposite to these sides are also equal, that is,

  $m\angle 5=m\angle 6$

Also, sum of angles in triangle is CDE is equal to $180°$ .

  $\begin{array}{l}m\angle 4+m\angle 5+m\angle 6=180°\\ 2k+m\angle 5+m\angle 5=180°\\ 2m\angle 5=180°-2k\\ m\angle 5=90°-k\end{array}$

Line AD is a straight line so,

  $\begin{array}{l}m\angle 2+m\angle 5+m\angle 7=180°\\ k+\left(90°-k\right)+m\angle 7=180°\\ m\angle 7=90°\end{array}$

Therefore, the measure of $\angle 7$ is $90°$ .