Question

1. in the rhombus, m∠3 = 80. find m∠2. 50 40 160 80

Explanation:

Step1: Recall rhombus properties

In a rhombus, the diagonals bisect the angles of the rhombus and the adjacent - sides are equal. Also, the sum of the interior angles of a triangle is 180°. In the triangle formed within the rhombus, if we consider the triangle with angles ∠1, ∠2, and ∠3, and we know that the diagonals of a rhombus are perpendicular bisectors of each other.

Step2: Use angle - sum property of a triangle

Let's assume the triangle formed by the diagonals and the sides of the rhombus. We know that one of the angles of the triangle is ∠3 = 80°. Since the diagonals of a rhombus are perpendicular to each other, the third angle of the triangle (the right - angle) is 90°. Let the third angle be ∠x = 90°.
We use the angle - sum property of a triangle: ∠1+∠2 + ∠3=180°. But in a rhombus, the diagonal bisects the angles of the rhombus. In the triangle formed by the diagonal and the sides of the rhombus, we know that ∠2 and ∠3 are related. Since the diagonals of a rhombus are perpendicular, in the right - triangle formed by the diagonal and the sides of the rhombus, if one non - right angle is ∠3 = 80°, then we can find ∠2.
We know that in a right - triangle, if one angle is ∠3 = 80° and the right - angle is 90°, and we want to find ∠2. Using the angle - sum property of a triangle (the sum of the interior angles of a triangle is 180°), we have ∠2=180°−90° - 80° = 10° is wrong.
The correct way: In a rhombus, the diagonals bisect the angles of the rhombus. The adjacent angles of a rhombus are supplementary. Let's consider the fact that the diagonal bisects the angles. In the triangle formed by the diagonal and the sides of the rhombus, since the diagonals of a rhombus are perpendicular, and we know that the opposite angles of a rhombus are equal and the adjacent angles are supplementary.
The diagonals of a rhombus bisect the angles of the rhombus. In the triangle formed by the diagonal and the sides of the rhombus, if ∠3 = 80°, and the diagonal bisects the angle of the rhombus, and the diagonals are perpendicular.
We know that the sum of the interior angles of a triangle is 180°. In the right - triangle formed by the diagonal and the sides of the rhombus, if one non - right angle is related to ∠3. Since the diagonal bisects the angle of the rhombus, and the diagonals are perpendicular, we know that ∠2 = 50°. Because if we consider the triangle formed by the diagonal and the sides of the rhombus, and we know that one angle of this triangle is 90° (diagonals are perpendicular) and ∠3 = 80° is part of the angle of the rhombus that is bisected by the diagonal.
Let's assume the triangle formed by the diagonal and two sides of the rhombus. The sum of the interior angles of a triangle is 180°. We know that one angle is 90° (diagonal - diagonal intersection). If we assume the angle at the vertex of the rhombus is 2∠3, and we know that in the right - triangle formed by the diagonal and the sides of the rhombus, ∠2=180°−90° - 40°=50° (because the diagonal bisects the angle of the rhombus and the angle related to ∠3 in the right - triangle is half of the angle of the rhombus at that vertex).

Answer:

50