Finding x:
We know that the diagonals of a rhombus bisect its angles
So, since US is a diagonal of the given rhombus:
∠RUS = ∠TUS
10x - 23 = 3x + 19 [replacing the given values of the angles]
7x - 23 = 19 [subtracting 3x from both sides]
7x = 42 [adding 23 on both sides]
x = 6 [dividing both sides by 7]
Finding ∠RUT:
We can see that:
∠RUT = ∠RUS + ∠TUS
Since we are given the values of ∠RUS and ∠TUS:
∠RUT = (10x - 23) + (3x + 19)
∠RUT = 13x - 4
We know that x = 6:
∠RUT = 13(6)- 4
∠RUT = 74°
