Question

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

Explanation:

Step1: Properties of rhombus

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

Step2: Find \(m\angle1\)

Since the adjacent angles of a rhombus are supplementary and the diagonals bisect the vertex angles. In a rhombus, the opposite angles are equal. Also, the diagonals bisect the angles. For the given rhombus, \(\angle1 = 90^{\circ}\) (because the diagonals of a rhombus are perpendicular to each other).

Step3: Find \(m\angle2\)

We know that in a right - triangle (formed by the diagonals of the rhombus), if one of the non - right angles is given as \(41^{\circ}\). Using the property that the sum of angles in a triangle is \(180^{\circ}\). In the right - triangle, \(m\angle2= 49^{\circ}\) (\(180-(90 + 41)=49\)).

Step4: Find \(m\angle3\)

\(\angle3=\angle1 = 90^{\circ}\) (opposite angles of a rhombus are equal and the diagonals bisect the angles, and the adjacent right - angled triangles are congruent).

Step5: Find \(m\angle4\)

\(\angle4=\angle2 = 49^{\circ}\) (opposite angles of a rhombus are equal and the diagonals bisect the angles, and the adjacent triangles are congruent).

Answer:

\(m\angle1 = 90^{\circ}\), \(m\angle2=49^{\circ}\), \(m\angle3 = 90^{\circ}\), \(m\angle4=49^{\circ}\)