If parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
Therefore, ∠ 2 and ∠ 3 are congruent and their measures are equal.
m ∠ 2=m ∠ 3
We already know that m ∠ 2= 40, so m ∠ 3= 40 as well. By the definition of a rectangle, we know that ABCD has four right angles. Therefore, the measure of m ∠ BCD is 90.
m ∠ BCD= 90
With the Angle Addition Postulate we can express m ∠ BCD as a sum of m ∠ 3 and m ∠ 4.
m ∠ 3 + m ∠ 4 = m ∠ BCD
⇕
m ∠ 3 + m ∠ 4 = 90
We know that m ∠ 3= 40. We will substitute it into the equation above to find the measure of ∠ 4.
40 + m ∠ 4 = 90
Let's solve it!
Because our quadrilateral is a rectangle, its diagonals are congruent and bisect each other. Therefore, the triangle formed by ∠ 4, ∠ 5, and ∠ 6 is an isosceles triangle, and by the definition of an isosceles triangle, the measures of ∠ 6 and ∠ 4 are equal.
m ∠ 6 = m ∠ 4
⇕
m ∠ 6 = 50
