Chapter 41, Problem 30A

To determine

(a)

The value of $\angle 1$ and verify the obtained solution.

Answer to Problem 30A

The value of $\angle 1$ is $24°$.

Explanation of Solution

Given:

Total angle is $180°$.

Calculation:

  

From Figure $1$, the value of $\angle 1=y$, $\angle 2=2y$, $\angle 3=3y-20°$ and $\angle 4=4y-40°$ respectively.

The sum of all angles is $180°$ and is expressed as below:

  $\angle 1+\angle 2+\angle 3+\angle 4=180°$.

From Figure $1$, the equation is observed as follows:

  $y+2y+3y-20°+4y-40°=180°$   ...... (1)

The value of $y$ is calculated by following steps:

  $\begin{array}{c}y+2y+3y-20°+4y-40°=180°\\ 10y-60°=180°\\ y=\frac{180°+60°}{10}\\ y=24°\end{array}$

Therefore, the value of $\angle 1$ is $24°$.

Substitute $24°$ for $y$ in equation (1) to verify the obtained solution as follows:

  $\begin{array}{c}y+2y+3y-20°+4y-40°=180°\\ 24°+\left(2\times 24°\right)+\left(3\times 24°\right)-20°+\left(4\times 24°\right)-40°=180°\\ 144°+36°=180°\\ 180°=180°\end{array}$

Thus, the value of $\angle 1$ is $24°$.

Conclusion:

The value of $\angle 1$ is $24°$.

To determine

(b)

The value of $\angle 2$ and verify the obtained solution.

Answer to Problem 30A

The value of $\angle 2$ is $48°$.

Explanation of Solution

Given:

Total angle is $180°$ and $\angle 2$ is $2y$.

Calculation:

From Figure $1$, the value of $\angle 2$ is calculated as follows:

  $\begin{array}{c}2y=2\times 24°\\ =48°\end{array}$

Therefore, the value of $\angle 2$ is $48°$.

Substitute $48°$ for $2y$ and $24°$ for $y$ in equation (1) to verify the obtained solution as follows:

  $\begin{array}{c}y+2y+3y-20°+4y-40°=180°\\ 24°+\left(48°\right)+\left(3\times 24°\right)-20°+\left(4\times 24°\right)-40°=180°\\ 144°+36°=180°\\ 180°=180°\end{array}$

Thus, the value of $\angle 2$ is $48°$.

Conclusion:

The value of $\angle 2$ is $48°$.

To determine

(c)

The value of $\angle 3$ and verify the obtained solution.

Answer to Problem 30A

The value of $\angle 3$ is $52°$.

Explanation of Solution

Given:

Total angle is $180°$ and $\angle 3$ is $3y-20°$.

Calculation:

From Figure $1$, the value of $\angle 3$ is calculated as follows:

Substitute $24°$ for $y$ in $3y-20°$ as below:

  $\begin{array}{c}3y-20°=\left(3\times 24°\right)-20°\\ =72°-20°\\ =52°\end{array}$

Therefore, the value of $\angle 3$ is $52°$.

Substitute $52°$ for $3y-20°$ and $24°$ for $y$ in equation (1) to verify the obtained solution as follows:

  $\begin{array}{c}y+2y+3y-20°+4y-40°=180°\\ 24°+\left(48°\right)+52°+\left(4\times 24°\right)-40°=180°\\ 144°+36°=180°\\ 180°=180°\end{array}$

Thus, the value of $\angle 3$ is $52°$.

Conclusion:

The value of $\angle 3$ is $52°$.

To determine

(d)

The value of $\angle 4$ and verify the obtained solution.

Answer to Problem 30A

The value of $\angle 4$ is $56°$.

Explanation of Solution

Given:

Total angle is $180°$ and $\angle 4$ is $4y-40°$.

Calculation:

From Figure $1$, the value of $\angle 4$ is calculated as follows:

  $\begin{array}{c}4y-40°=4\times 24°-40°\\ =96°-40°\\ =56°\end{array}$

Therefore, the value of $\angle 4$ is $56°$.

Substitute $56°$ for $4y-40°$ and $24°$ for $y$ in equation (1) to verify the obtained solution as follows:

  $\begin{array}{c}y+2y+3y-20°+4y-40°=180°\\ 24°+\left(48°\right)+\left(3\times 24°\right)-20°+56°=180°\\ 144°+36°=180°\\ 180°=180°\end{array}$

Thus, the value of $\angle 4$ is $56°$.

Conclusion:

The value of $\angle 4$ is $56°$.