Question

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:

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