Exercise 27 Page 509

Let's analyze the given quadrilateral so that we can find the measure of ∠ 7.

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

Therefore, ∠ 2 and ∠ 3 are congruent. Since the measure of ∠ 2 is given, we can use it. m ∠ 3=m ∠ 2 ⇔ m ∠ 3 = 40 Because our quadrilateral is a rectangle, its diagonals are congruent and bisect each other. Therefore, the triangle formed by ∠ 3, ∠ 7, and ∠ 8 is an isosceles triangle. By the definition of an isosceles triangle, we know that m ∠ 7 and m ∠ 3 are congruent. m ∠ 7=m ∠ 3 ⇔ m ∠ 7 = 40 Therefore, the measure of ∠ 7 is 40.