find the measures of the numbered angles in the rhombus.\nm∠1=□°

Explanation:

Step1: Recall rhombus diagonal properties

In a rhombus, diagonals are perpendicular bisectors of each other and bisect the angles of the rhombus. Also, the diagonals form right triangles with the sides. In a right triangle, the sum of angles is (180^\circ), and one angle is (90^\circ) (from perpendicular diagonals).

Step2: Calculate (m\angle1)

We know one angle in the right triangle (formed by diagonals) is (31^\circ), and (\angle2 = 90^\circ) (since diagonals are perpendicular). So, in the right triangle, (m\angle1+ 31^\circ+ 90^\circ=180^\circ)? Wait, no, actually, the diagonal bisects the angle, and the triangle has angles: (\angle1), (31^\circ), and (90^\circ)? Wait, no, correction: In a rhombus, the diagonals bisect the vertex angles. Also, the diagonals are perpendicular, so (\angle2 = 90^\circ). The triangle with angle (31^\circ), (\angle1), and (\angle2 = 90^\circ) – wait, actually, the angle adjacent to (31^\circ) in the right triangle: since the diagonal bisects the angle, and the triangle is right - angled, (m\angle1 = 90^\circ - 31^\circ). Calculating (90 - 31=59). So (m\angle1 = 59^\circ).

Answer:

(59)