Given parallel lines AXB and DYZF, and angles X3 = 55^\[\ and Y1 = 45^\[\ (as defined in the solution), find Y2, Z1, and Z3.

Here's how to calculate the required angles:

Given:

• Line AXB is parallel to line DYZF ($AXB \parallel DYZF$).
• $\hat{X}_3 = \angle YXZ = 55^\circ$.
• $\hat{Y}_1 = \angle XYD = 45^\circ$.

We need to find $\hat{Y}_2$, $\hat{Z}_1$, and $\hat{Z}_3$.

Step 1: Find $\angle AXY$.
Since $AXB \parallel DYZF$, $\angle AXY$ and $\hat{Y}_1$ are alternate interior angles.

$$\angle AXY = \hat{Y}_1 = 45^\circ$$