We can find the measure of angles RQS and TQS by applying the theorem below:
The sum of angles on a straight line is 180 degrees
Given:
[tex]\begin{gathered} m\angle RQS=(20x+4)^0 \\ m\angle TQS=(15x+1)^0 \end{gathered}[/tex]Applying the theorem, we have:
[tex](20x+4)^0+(15x+1)^0=180^0\text{ (angles on a straight line)}[/tex]Simplifying and solving for x:
[tex]\begin{gathered} 20x\text{ + 4 + 15x + 1 =180} \\ \text{Collect like terms} \\ 20x\text{ + 15x + 5 =180} \\ 35x\text{ = 180-5} \\ 35x\text{ = 175} \\ Divide\text{ both sides by 35} \\ \frac{35x}{35}\text{ = }\frac{175}{35} \\ x\text{ = 5} \end{gathered}[/tex]When we substitute the value of x , we can find the required angles.
Hence:
[tex]\begin{gathered} m\angle RQS=(20x+4)^0 \\ =\text{ 20}\times5\text{ + 4} \\ =\text{ 100 + 4} \\ =104^0 \end{gathered}[/tex]Answer: measure of angle RQS = 104 degrees
[tex]\begin{gathered} m\angle TQS=(15x+1)^0 \\ =\text{ 15 }\times\text{ 5 + 1} \\ =\text{ 75 + 1} \\ =76^0 \end{gathered}[/tex]Answer: measure of angle TQS = 76 degrees