for the rhombus below, find the measures of ∠1, ∠2, ∠3, and ∠4.

Explanation:

Step1: Recall rhombus properties

In a rhombus, the diagonals are perpendicular bisectors of each other and bisect the angles of the rhombus.

Step2: Find measure of ∠1

Since the diagonals of a rhombus are perpendicular, if one of the angles formed by the diagonals is 90°. Given an angle adjacent to ∠1 in the right - triangle formed by the diagonals is 42°. Using the fact that the sum of angles in a triangle is 180°, in the right - triangle with one angle 90° and another 42°, we have $\angle1=90^{\circ}- 42^{\circ}=48^{\circ}$.

Step3: Find measure of ∠2

The diagonal of a rhombus bisects the angle of the rhombus. ∠1 and ∠2 are equal because the diagonal bisects the vertex angle of the rhombus. So $\angle2 = 48^{\circ}$.

Step4: Find measure of ∠3

Opposite angles of a rhombus are equal. The angle opposite to the angle containing ∠1 and ∠2 has the same measure. Also, the diagonal bisects this opposite angle. So $\angle3=48^{\circ}$.

Step5: Find measure of ∠4

∠4 is equal to the given 42° angle because the diagonal bisects the angle of the rhombus. So $\angle4 = 42^{\circ}$.

Answer:

$m\angle1 = 48^{\circ}$ $m\angle2 = 48^{\circ}$ $m\angle3 = 48^{\circ}$ $m\angle4 = 42^{\circ}$