The lst time I did geometry like this was 50 years ago, so I'm a bit vague about the names of geometrical proofs and theorems. I can only give you a hint about what to look for yourself.
You have a triangle with two angles, 23 and 48.
Let the third angle be T .
23 + 48 + T = 180 (sum of angles of a triangle add up to 180)
T = 109
x = T (opposite angles ???)
x = 109
In the quadrilateral,
81 + z = 180 (because of the two parallel lines)
z = 99
Let the angle opposite to y be T2
T2 + x + z + 81 = 360 (sum of internal angles of a quadrilateral add up to 360)
T2 + 109 + 99 + 81 = 360
T2 + 289 = 360
T2 = 71
y = T2 (opposite angles ???)
y = 71
The three angles are: x = 109, y = 71, z = 99