find the measures of all the angles in the rectangle.

Explanation:

Step1: Use properties of rectangle

In a rectangle, each angle is (90^{\circ}). So (\angle2 = \angle5=90^{\circ}).

Step2: Find (\angle1)

In a right - triangle (formed by the side of the rectangle and the diagonal), if one angle is (22^{\circ}), and the right - angle is (90^{\circ}). Using the angle - sum property of a triangle ((\angle1+22^{\circ}+90^{\circ}=180^{\circ})), we get (\angle1 = 90^{\circ}-22^{\circ}=68^{\circ}).

Step3: Use alternate interior angles (rectangle has parallel sides)

Since the opposite sides of a rectangle are parallel, (\angle3=\angle1 = 68^{\circ}) (alternate interior angles).

Step4: Use angle - sum property for (\angle4)

In the right - triangle (with (\angle4)), (\angle4+ \angle3+90^{\circ}=180^{\circ}). Substituting (\angle3 = 68^{\circ}), we get (\angle4=90^{\circ}-\angle3=22^{\circ}).

Answer:

(\angle1 = 68^{\circ}), (\angle2 = 90^{\circ}), (\angle3 = 68^{\circ}), (\angle4 = 22^{\circ}), (\angle5 = 90^{\circ})