Chapter 1, Problem 29RP

To determine

To Find: measure of missing angle from the given figure.

Answer to Problem 29RP

Measure of angle $1$= $39°50\text{'}44"$

Explanation of Solution

Given information: In the given figure a straight line is given where measurement of two angles is given as $69°29\text{'},70°40\text{'}16"\text{ and m}\angle \text{1}$ is to be found.

Calculation:

We know that sum of all the angles on a straight line add up to $180°$

So, here $\text{ 69}°\text{29' + 70}°\text{40'16"+ }m\angle 1=180°\mathrm{......................................}(\text{i})$

Converting given angles completely in degrees using DMS(Degree ,minute, second) relation according to which to convert minute(‘) into degree we divide by $60$ and to convert seconds(“) into degrees we divide by $3600$

Considering $69°29\text{'}$ ,it can be written as sum of degrees and minutes as:

  $\begin{array}{c}\text{69}°\text{29'= 69}°\text{+29'}\\ \text{= 69}°\text{+}{\left(\frac{29}{60}\right)}^{°}\\ =\text{ 69}°\text{ + 0}\text{.4833}°\\ \text{=69}\text{.4833}°\mathrm{............................................................}\text{(ii)}\end{array}$

And $70°40\text{'}16"$ can be written as sum of degrees and minutes as:

  $\begin{array}{c}\text{70}°40\text{'}16"=70°\text{ + 40' + 16"}\\ \text{= 70}°\text{ + }{\left(\frac{40}{60}\right)}^{°}\text{ + }{\left(\frac{16}{3600}\right)}^{°}\\ \text{= 70}°\text{ + 0}\text{.6667}°\text{ + 0}\text{.0044}°\\ \text{= 70}\text{.6711}°\mathrm{.....................................................................}\text{(iii)}\end{array}$

Substituting values from (ii) ,(iii) into (i) we get :

  $\begin{array}{l}69.4833°+70.6711°\text{ +}m\angle \text{1= 180}°\\ \Rightarrow \text{ 140}\text{.1544}°\text{ + }m\angle \text{1= 180}°\\ \Rightarrow m\angle \text{1 = 180}°\text{ - 140}\text{.1544}°\\ \Rightarrow m\angle \text{1= 39}\text{.8456}°\mathrm{.......................................}\text{(iv)}\end{array}$

  $$

Now converting $39.8456°$ into degrees ,minutes and seconds by multiplying decimal part first by $60$ to get minutes and then again multiplying the decimal part by $60$ to get seconds as shown below;

Minutes $\text{= 0}\text{.8456}°\times \text{ 60 =50}\text{.736'}$ = $50\text{'}$

Seconds = $$

  $\text{0}\text{.736' }\times \text{ 60 = 44}\text{.16 " or 44" }\text{.}$

Plugging in values of minutes and seconds in (iv) we get:

  $m\angle \text{1 = 39}°\text{ 50' 44"}$

Conclusion:

  $\text{Measure of angle 1 or m}\angle \text{1 = 39}°\text{ 50' 44"}$