The given parallelogram is a rhombus, as it has four congruent sides. Let's find the measure of each numbered angle one at a time.
Measure of ∠ 3
If a parallelogram is a rhombus, then its diagonals are perpendicular.
Therefore, we can conclude that ∠ 3 is a right angle and has a measure of 90.
Measure of ∠2
Notice that ∠ 2 and the angle of measure 32 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 the angles are congruent.
∠ 2 ≅ 32
By the definition of congruent angles, we know that their measures are equal. We also know that m ∠ 2 = 32.
Measure of ∠ 1
To find the measure of ∠ 1 we need to study the triangle formed by ∠ 1, ∠ 2, and the angle where the diagonals intersect. Again we can make use of the fact that the diagonals are perpendicular. Therefore, the triangle's third angle is right.
By the Triangle Angle-Sum Theorem we can conclude that their measures add to 180. We already know that m ∠ 2 =32, and that the angle formed by the diagonals is right and its measure is 90.
m ∠ 1 + 32 + 90=180
Let's solve the equation!
