Exercise 19 Page 426

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

First, we must find the value of x. By the definition of a rectangle, we know that WXYZ has four right angles. Therefore, the measure of m ∠ WXY is 90. m ∠ WXY= 90With the Angle Addition Postulate we can express m ∠ WXY as a sum of m ∠ ZXY and m ∠ ZXW. m ∠ ZXY + m ∠ ZXW = m ∠ WXY ⇕ m ∠ ZXY + m ∠ ZXW = 90 We are given that m ∠ XZW= x-11. We will substitute this expressions into our equation. m ∠ ZXY + x-11 = 90 Because our quadrilateral is a rectangle, both pairs of opposite sides are parallel. Recall the following theorem.

Notice that by this theorem, ∠ ZXY and ∠ WZX are alternate interior angles, with ZX as the transversal. Therefore, ∠ ZXY and ∠ WZX are congruent. An expression for the measure of ∠ WZX is given as x-9, and we can substitute it into the expression. m ∠ ZXY&=m ∠ WZX &⇕ m ∠ ZXY&= x-9 Now that we have an expression for the measure of ∠ WZX, let's substitute it back into the previous equation. x-9 + x-11 = 90 Let's solve it!

Finally, we can substitute x=55 into m∠ ZXY= x-9 equation to find the measure of the angle.