Answer:
X = 34°
Step-by-step explanation:
In order to solve this question we will need to know the following, when two lines intersect the opposite angles at that point are equal to each other, that the sum of all angles of a triangle is 180 degrees, and that the angle of a straight line is equal to 180 degrees.
Frist we will have to notice that ∠TSU is equal to ∠PSR, because they are opposite to each other (If you would like to understand this more you can research on a topic call "vertical angels"). Now it is important to understand the following.
∠PSR = ∠PSQ + QSR
Since ∠PSR = ∠TSU = 4x, ∠QSR = 3x ⇒
⇒ ∠PSR = ∠PSQ + QSR
4x = ∠PSQ + 3x
4x - 3x = ∠PSQ
x = ∠PSQ
Now, since the sum of all angles in a triangle is equal to 180 we can now say the following:
∠QPS + ∠PSQ + ∠PQS = 180°
In order to find out what PQS is equal to we need to use our knowledge about the angel of a strait line (the angel of a strait line is equal to 180°).
So now we can say the following:
180° = ∠PQS + SQR
∠PQS = 180° - ∠SQR
Since we know that ∠SQR = 68° ⇒
⇒∠PQS = 180° - ∠SQR
∠PQS = 180° - 68°
∠PQS = 112°
Since ∠QPS = x, PSQ = x, and ∠PQS = 112° ⇒
⇒ ∠QPS + ∠PSQ + ∠PQS = 180°
x + x + 112° = 180°
2x = 180° - 112°
2x = 68°
x = 34°
