for the rhombus below, find the measures of $\\angle 1$, $\\angle 2$, $\\angle 3$, and $\\angle…

Explanation:

Step1: Recall Rhombus Diagonal Properties

In a rhombus, diagonals bisect the angles. Also, diagonals are perpendicular bisectors of each other, so they form right angles? Wait, no, diagonals bisect the vertex angles and are perpendicular. Wait, the triangles formed by the diagonals are congruent. Also, alternate interior angles? Wait, in a rhombus, opposite sides are parallel, so alternate interior angles are equal. Also, diagonals bisect the angles. So the given angle is (47^\circ), so (\angle 1) should be equal to (47^\circ) because of alternate interior angles (since sides are parallel) or because diagonals bisect angles? Wait, let's think again.

In a rhombus, the diagonals bisect the vertex angles. Also, adjacent angles are supplementary, but here we have triangles. Wait, the diagonals of a rhombus bisect the angles, so if one angle is (47^\circ), then (\angle 1 = 47^\circ). Then, since the diagonals are perpendicular? Wait, no, diagonals of a rhombus are perpendicular bisectors, so the triangles formed are right triangles? Wait, no, in a rhombus, diagonals are perpendicular, so the angles between diagonals are (90^\circ)? Wait, no, the diagonals intersect at right angles. So in each triangle formed by the diagonals, we have a right triangle? Wait, maybe not. Let's look at the angles.

Wait, the given angle is (47^\circ), so (\angle 1 = 47^\circ) (alternate interior angles, since sides are parallel). Then, (\angle 2) and (\angle 4): since diagonals bisect the angles, and in a rhombus, adjacent angles are supplementary. Wait, maybe (\angle 2 = 90^\circ - 47^\circ = 43^\circ)? No, wait, maybe (\angle 3) is (90^\circ)? No, that doesn't make sense. Wait, let's recall: in a rhombus, diagonals bisect the angles, so each diagonal splits the angle into two equal parts. Also, the diagonals are perpendicular, so the angles between the diagonals are (90^\circ). Wait, no, the diagonals intersect at right angles. So in the triangle, if one angle is (47^\circ), then the other angle (like (\angle 2)) would be (90^\circ - 47^\circ = 43^\circ)? Wait, maybe I'm overcomplicating.

Wait, the key properties:

1. In a rhombus, opposite sides are parallel, so alternate interior angles are equal. So (\angle 1 = 47^\circ) (since the sides are parallel, the transversal is the diagonal, so alternate interior angles are equal).

2. Diagonals of a rhombus bisect the angles, so (\angle 2 = \angle 4), and (\angle 1 = \angle 3)? Wait, no, maybe (\angle 3) is equal to (\angle 1)? Wait, no, let's think about the right angles. Wait, diagonals of a rhombus are perpendicular, so the angles between the diagonals are (90^\circ). Wait, no, the diagonals intersect at right angles, so in each triangle, we have a right angle? Wait, no, the diagonals intersect at (90^\circ), so the angle between the diagonals is (90^\circ). So in the triangle, if one angle is (47^\circ), then the other angle (non-right) is (43^\circ). Wait, maybe (\angle 2 = 43^\circ), (\angle 3 = 90^\circ)? No, that can't be. Wait, maybe (\angle 3) is (90^\circ) because diagonals are perpendicular? Wait, no, the diagonals intersect at right angles, so the angle between the diagonals is (90^\circ), so (\angle 3 = 90^\circ)? No, that's not right.

Wait, let's start over.

• In a rhombus, diagonals bisect the vertex angles. So each diagonal splits the angle into two equal parts.

• Also, the diagonals are perpendicular (they intersect at (90^\circ)).

So, given that one angle is (47^\circ) (let's say at the top right), then the diagonal bisects that angle into two (47^\circ) angles? No, wait, the diagonal would split the angle into two equal parts. Wait, maybe the given angle is (47^\circ), so (\angle 1 = 47^\circ) (because of alternate interior angles, since the sides are parallel, the diagonal is a transversal, so alternate interior angles are equal). Then, since the diagonals are perpendicular, (\angle 3 = 90^\circ)? No, that's not. Wait, maybe (\angle 2 = 43^\circ), (\angle 4 = 43^\circ), and (\angle 3 = 90^\circ)? No, I'm confused.

Wait, let's look at the diagram. The rhombus has diagonals intersecting, forming four triangles. One of the angles in the top triangle is (47^\circ). So (\angle 1) is equal to that (47^\circ) angle (alternate interior angles, since the sides are parallel). Then, (\angle 2) and (\angle 4) are equal, and since the diagonals are perpendicular, the triangle is a right triangle? Wait, no, diagonals of a rhombus are perpendicular, so the angle between the diagonals is (90^\circ), so (\angle 3 = 90^\circ)? No, that's the angle between the diagonals. Wait, maybe (\angle 3) is (90^\circ), (\angle 1 = 47^\circ), (\angle 2 = 43^\circ), (\angle 4 = 43^\circ). Let's check:

In a right triangle, the sum of angles is (180^\circ). So (47^\circ + 43^\circ + 90^\circ = 180^\circ), which works. So that makes sense.

So:

• (\angle 1 = 47^\circ) (alternate interior angles, since sides are parallel, diagonal is transversal)

• (\angle 2 = 90^\circ - 47^\circ = 43^\circ) (since diagonals are perpendicular, so the triangle is right-angled, so angles sum to (180^\circ))

• (\angle 3 = 90^\circ) (diagonals of rhombus are perpendicular, so angle between diagonals is (90^\circ))

• (\angle 4 = 43^\circ) (since diagonals bisect angles, and (\angle 2 = \angle 4))

Wait, but why is (\angle 3 = 90^\circ)? Because diagonals of a rhombus are perpendicular, so they intersect at (90^\circ), so (\angle 3) is the right angle.

Let me confirm the properties:

1. All sides of a rhombus are equal.

2. Opposite sides are parallel.

3. Opposite angles are equal.

4. Diagonals bisect each other at right angles (perpendicular bisectors).

5. Diagonals bisect the angles of the rhombus.

So, from property 4, diagonals intersect at right angles, so (\angle 3 = 90^\circ) (since it's the angle between the diagonals).

From property 5, diagonals bisect the angles, so the (47^\circ) angle is bisected? Wait, no, the given angle is (47^\circ), which is one of the angles formed by the diagonal and the side. So since opposite sides are parallel, (\angle 1 = 47^\circ) (alternate interior angles). Then, in the right triangle (since diagonals are perpendicular), the angles are (47^\circ), (90^\circ), and (43^\circ). So (\angle 2 = 43^\circ), (\angle 4 = 43^\circ) (since diagonals bisect angles, so (\angle 2 = \angle 4)).

So putting it all together:

• (m\angle 1 = 47^\circ)

• (m\angle 2 = 43^\circ)

• (m\angle 3 = 90^\circ)

• (m\angle 4 = 43^\circ)

Wait, but why is (\angle 3 = 90^\circ)? Because diagonals of a rhombus are perpendicular, so the angle between them is (90^\circ), so (\angle 3) is that right angle.

Let me check with another approach. In a rhombus, adjacent angles are supplementary. So if one angle is (2 \times 47^\circ = 94^\circ), then the adjacent angle is (180^\circ - 94^\circ = 86^\circ), and the diagonal bisects that into (43^\circ) each. So (\angle 2 = 43^\circ), (\angle 4 = 43^\circ), (\angle 1 = 47^\circ), and (\angle 3 = 90^\circ) (since diagonals are perpendicular).

Yes, that makes sense. So:

Step1: Determine (\angle 1)

In a rhombus, opposite sides are parallel. The diagonal acts as a transversal, so alternate interior angles are equal. Thus, (m\angle 1 = 47^\circ).

Step2: Determine (\angle 3)

Diagonals of a rhombus are perpendicular (intersect at (90^\circ)). Thus, (m\angle 3 = 90^\circ).

Step3: Determine (\angle 2) and (\angle 4)

In a triangle, the sum of angles is (180^\circ). For the triangle with angles (\angle 1), (\angle 2), and (\angle 3): [ m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ ] Substitute (m\angle 1 = 47^\circ) and (m\angle 3 = 90^\circ): [ 47^\circ + m\angle 2 + 90^\circ = 180^\circ ] [ m\angle 2 = 180^\circ - 47^\circ - 90^\circ = 43^\circ ] Since diagonals bisect angles, (m\angle 4 = m\angle 2 = 43^\circ).

Answer:

(m\angle 1 = \boxed{47})°, (m\angle 2 = \boxed{43})°, (m\angle 3 = \boxed{90})°, (m\angle 4 = \boxed{43})°