Question

find the measures of the numbered angles in the rhombus.

Explanation:

Step1: Recall Rhombus Properties

In a rhombus, diagonals are perpendicular bisectors, so \(\angle 2 = 90^\circ\). Also, diagonals bisect the angles of the rhombus, and opposite angles are equal, alternate interior angles are equal.

Step2: Find \(\angle 3\)

The given angle is \(31^\circ\), and by the property of rhombus diagonals bisecting angles, \(\angle 3 = 31^\circ\) (alternate interior angles or angle - bisecting property).

Step3: Find \(\angle 1\)

In the right - triangle formed (since \(\angle 2 = 90^\circ\)), we know that the sum of angles in a triangle is \(180^\circ\). So \(\angle 1=90^\circ - 31^\circ=59^\circ\).

Step4: Find \(\angle 4\)

By the property of rhombus diagonals bisecting angles, \(\angle 4=\angle 1 = 59^\circ\) (or we can use the right - triangle angle sum again, \(\angle 4 = 90^\circ-31^\circ = 59^\circ\)).

Answer:

\(\angle 1 = 59^\circ\), \(\angle 2=90^\circ\), \(\angle 3 = 31^\circ\), \(\angle 4 = 59^\circ\)