The given parallelogram is a rectangle, as its opposite sides are parallel and one of its angles is right. Let's find the measure of each numbered angle one at a time.
Measure of ∠ 1
We see that ∠ 1 and the angle with measure 56^(∘) form a linear pair. Therefore, we can conclude that their measures add to 180.
m ∠ 1 + 56 =180 ⇔ m ∠ 1 =124
Measure of ∠ 3
If a parallelogram is a rectangle, then we know that its diagonals are congruent. We also know that for parallelograms it is true that the diagonals bisect each other. Therefore, the four segments that meet inside the rectangle must be congruent.
Now we see that the triangle with ∠ 3 and the angle with measure 56^(∘) is an isosceles triangle. The Isosceles Triangle Theorem tells us that the angles opposite the congruent sides are congruent. Let's label the third angle ∠ 3'. By the Triangle Angle-Sum Theorem we can conclude that the measures of the three angles add to 180.
m ∠ 3 + m ∠ 3' + 56=180
The base angles, ∠ 3 and ∠ 3', are congruent angles. By the definition of congruent angles, we know that their measures are equal. Let's use this to solve the equation for m∠ 3.
In a rectangle all angles are right and have the measure 90. Therefore, m∠ 2 and m∠ 3 must add to 90.
m∠ 2 + m∠ 3=90
Let's substitute 62 for m∠ 3 and solve for m ∠ 2.
m∠ 2 + 62=90 ⇔ m ∠ 2 = 28
Therefore, the measure of ∠ 2 is 28.
