Let's find the measure of each numbered angle in the given rhombus one at a time.
Measure of ∠ 1
In a rhombus then each diagonal bisects opposite angles.
Therefore, ∠ 1 and the angle of measure 57 are congruent. By the definition of congruent angles, we know that their measures are equal.
m ∠ 1 = 57
Measure of ∠ 2
Notice that ∠ 2 and the angle of measure 57 are alternate interior angles. Since a rhombus is a parallelogram, we know that its opposite sides are parallel. Therefore, by the Alternate Interior Angles Theorem these angles are congruent. By the definition of congruent angles, we know that their measures are equal.
m ∠ 2 = 57
Measure of ∠ 3
Notice that ∠ 1, ∠ 2, and ∠ 3 are three angles in a triangle. Therefore, by the Triangle Angle-Sum Theorem, their measures add up to 180.
m ∠ 1 + m ∠ 2 + m ∠ 3 = 180
We already know that m ∠ 1 = 57 and m ∠ 2 = 57. Let's substitute these values in our equation.
57 + 57 + m ∠ 3 = 180 ⇕ m ∠ 3 = 66
Therefore, the measure of ∠ 3 is 66.
