Chapter 1, Problem 31RP

a )

To determine

To Find : The measure of angle ABD to the nearest tenth of a degree

Answer to Problem 31RP

The measure of angle ABD to the nearest tenth of a degree = $44.5°$

Explanation of Solution

Given: In the given figure on line AC two adjacent angles $\angle ABD=(812x\text{)}°\text{ , }\angle DBC\text{ = (2469}x\text{)}°$ are marked

Concept Used: Two adjacent angles standing on a straight line form a linear pair which means they add up to $180°$

Calculation: As per the given figure $\angle ABD$ , $\angle DBC$ form a linear pair. So,

  $\angle \text{ABD + }\angle DBC\text{ = 180}°$ also $\angle ABD=(812x\text{)}°\text{ , }\angle DBC\text{ = (2469}x\text{)}°$ .Which means:

  $\begin{array}{l}\angle ABD\text{ + }\angle DBC\text{ = 180}°\\ \Rightarrow (812x\text{)}°\text{+ ( 2469}x\text{)}°\text{ = 180}°\\ \Rightarrow \text{ (3281}x\text{)}°\text{ = 180}°\\ \Rightarrow x\text{ = }\left(\frac{180°}{3281°}\right)\\ \\ \Rightarrow x\text{ = 0}\text{.05486}\end{array}$

(Taking value up to $5$ decimal places)

So, $\angle ABD\text{ = (812}x)°\Rightarrow \angle \text{ABD = (812 }\times \text{ 0}\text{.05486)}°\text{ = 44}\text{.54632}°$

Converting this value to the nearest tenth place we will get $\text{44}\text{.54632}°\text{ = 44}\text{.5}°$

Conclusion:

The measure of angle ABD to the nearest tenth of a degree = $44.5°$

b)

To determine

To Find: The measure of given angle to its nearest minute

Answer to Problem 31RP

The measure of given angle to its nearest minute $\text{= 44}°\text{ 33 '}$

Explanation of Solution

Given: In the given figure on line AC two adjacent angles $\angle ABD=(812x\text{)}°\text{ , }\angle DBC\text{ = (2469}x\text{)}°$ are marked

Concept Used: Two adjacent angles standing on a straight line form a linear pair which means they add up to $180°$

Calculation: As per the given figure $\angle ABD$ , $\angle DBC$ form a linear pair. So,

  $\angle \text{ABD + }\angle DBC\text{ = 180}°$ also $\angle ABD=(812x\text{)}°\text{ , }\angle DBC\text{ = (2469}x\text{)}°$ .Which means:

  $\begin{array}{l}\angle ABD\text{ + }\angle DBC\text{ = 180}°\\ \Rightarrow (812x\text{)}°\text{+ ( 2469}x\text{)}°\text{ = 180}°\\ \Rightarrow \text{ (3281}x\text{)}°\text{ = 180}°\\ \Rightarrow x\text{ = }\left(\frac{180°}{3281°}\right)\\ \\ \Rightarrow x\text{ = 0}\text{.05486}\end{array}$

(Taking value up to $5$ decimal places)

So, $\angle ABD\text{ = (812}x)°\Rightarrow \angle \text{ABD = (812 }\times \text{ 0}\text{.05486)}°\text{ = 44}\text{.54632}°$

Converting $\text{44}\text{.54632}°$ into minutes by multiplying decimal part by $60$

  $\text{0}\text{.54632 }°\times \text{ 60 = 32}\text{.7792 minutes}$

Writing it to the nearest minute we will get : $\text{44}°\text{ 33 '}$

Conclusion:

The measure of given angle to its nearest minute $\text{= 44}°\text{ 33 '}$