First, we need to find the coordinates of each vertex of triangle ABC. Let's assume that A is located at (a,b), B is located at (c,d), and C is located at (e,f). Since we know the measures of the angles, we can also determine the slopes of the lines connecting the vertices:
The slope of line AB is (d-b)/(c-a), which is equal to tan(55°).
The slope of line BC is (f-d)/(e-c), which is equal to tan(80°).
The slope of line AC is (f-b)/(e-a), which is equal to tan(55°-45°) = tan(10°).
Using these slopes, we can find the equations of the three lines and solve for the coordinates of the vertices:
Line AB: y-b = tan(55°)(x-a)
Line BC: y-d = tan(80°)(x-c)
Line AC: y-b = tan(10°)(x-a)
Solving these equations simultaneously, we get:
A = (b + (c-a)tan(55°), b + (c-a)tan(55°-45°))
B = (c + (f-d)/tan(80°), d + (f-d))
C = (e + (b-f)/tan(10°), f + (e-a)tan(10°))
Next, we need to apply the rotation and reflection to these vertices to find the corresponding vertices of triangle XYZ. The rotation by 90° counterclockwise about the origin transforms a point (x,y) into (-y,x), while the reflection across the line y=x transforms a point (x,y) into (y,x). So:
Vertex A of ABC is mapped to vertex X of XYZ: (a,b) → (-b,a) → (a,-b)
Vertex B of ABC is mapped to vertex Y of XYZ: (c,d) → (-d,c) → (c,d)
Vertex C of ABC is mapped to vertex Z of XYZ: (e,f) → (-f,e) → (e,f)
Now we need to find the measure of angle Y. Since we don't know the exact coordinates of the vertices of XYZ, we'll use the fact that the rotation by 90° counterclockwise about the origin preserves angles and the reflection across the line y=x changes the orientation of angles but not their measure. Therefore:
m∠Y = m∠ZOX, where O is the origin
m∠ZOX = 90° - m∠XOZ
m∠XOZ = m∠COA = 55° + 45° = 100°
Therefore, m∠Y = 90° - 100° = -10°, but since angles can't have negative measures, we add 360° to get m∠Y = 350°.
So the measure of angle Y in triangle XYZ is 350°.