Answer:
[tex]x = 34^\circ[/tex]
Step-by-step explanation:
Note that ∠TSU and ∠PSR are vertical angles. Hence:
[tex]m\angle TSU = m\angle PSR[/tex]
∠PSR is the sum of ∠PSQ and ∠QSR. Hence:
[tex]\displaystyle m\angle TSU = m\angle PSQ + m\angle QSR[/tex]
We know that ∠TSU measures 4x and ∠QSR measures 3x. Thus:
[tex](4x) = m\angle PSQ + (3x)[/tex]
Solve for ∠PSQ:
[tex]m\angle PSQ = x[/tex]
Next, ∠PQS and ∠RQS form a linear pair. Thus:
[tex]m\angle PQS + m\angle RQS = 180^\circ[/tex]
∠RQS measures 68°. Thus:
[tex]m\angle PQS +(68^\circ) = 180^\circ[/tex]
Solve for ∠PQS:
[tex]m\angle PQS = 112^\circ[/tex]
The interior angles of a triangle must total 180°. So, for ΔPQS:
[tex]\displaystyle m\angle SPQ + m\angle PQS + m\angle PSQ = 180^\circ[/tex]
Substitute in the known values:
[tex](x) + (112^\circ) + (x) = 180^\circ[/tex]
Simplify:
[tex]2x = 68^\circ[/tex]
And divide. Hence:
[tex]x = 34^\circ[/tex]
