Answer:
x is -4.
TRS is 89.
Step-by-step explanation:
So we know that QR bisects QRS. By the definition of an angle bisector, this means that the adjacent angles are equivalent. In other words:
[tex]\angle QRT=\angle TRS[/tex]
We are told that QRS is (9x+214). QRS is the sum of QRT and TRS. Thus:
[tex]\angle QRS=\angle QRT+\angle TRS[/tex]
And since we now that QRT and TRS are equivalent, substitute:
[tex]\angle QRS=\angle TRS+\angle TRS[/tex]
Combine like terms:
[tex]\angle QRS=2\angle TRS[/tex]
Substitute them for their equations:
[tex]9x+214=2(-9x+53)[/tex]
Now, solve for x. On the right, distribute:
[tex]9x+214=-18x+106[/tex]
Add 18x to both sides:
[tex]27x+214=106[/tex]
Subtract 214 from both sides:
[tex]27x=-108[/tex]
Divide both sides by 27:
[tex]x=-4[/tex]
So, the value of x is -4.
To find TRS, substitute x=-4 back into the equation. So:
[tex]\angle TRS=(-9(4))+53[/tex]
Multiply and add:
[tex]\angle TRS=36+53=89\textdegree[/tex]
And we're done!
