Exercise 29 Page 427

Let's analyze the given quadrilateral so that we can find the measure of ∠ 5.

Firstly, notice that ∠ 2 and ∠ 3 form alternate interior angles. Because our quadrilateral is a rectangle, both pairs of opposite sides are parallel. Recall the following theorem.

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 ∠ 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 m∠ 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 with angles m ∠ 4, m ∠ 5, and m ∠ 6 is an isosceles triangle. Because of the definition of an isosceles triangle, we know that m ∠ 6 = m ∠ 4. Recall the following theorem.

We can write an equation for the angles of the triangle. m ∠ 4 + m ∠ 5 + m ∠ 6 = 180 ⇕ 50 + m ∠ 5 + 50 = 180 Finally, let's solve it to find m ∠ 5.