Let's start by analyzing the given quadrilateral so that we can find the measure of ∠ YXZ.
First, we must find the value of x. By the definition of a rectangle, we know that WXYZ has four right angles. This means that the measure of m ∠ YZW is 90.
m ∠ YZW= 90Next, using the Angle Addition Postulate, we can express m ∠ YZW as a sum of m ∠ XZY and m ∠ XZW.
m ∠ XZY + m ∠ XZW = m ∠ YZW
⇕
m ∠ XZY + m ∠ XZW = 90
Now, we are given that m ∠ XZY= 3x+6, and m ∠ XZW= 5x-12. We will substitute these expressions into our equation.
3x+6 + 5x-12 = 90
Let's solve the equation for x.
Now we want to find m ∠ YXZ. Notice that ∠ XZW and ∠ YXZ form alternate interior angles. Because our quadrilateral is a rectangle, both pairs of opposite sides are parallel. Therefore, by the Alternate Interior Angles Theorem, ∠ YXZ and ∠ XZW are congruent. This means that their measures are equal.
m ∠ YXZ=m ∠ XZW
We already know that m ∠ XZW=5x-12. Substituting this in the above equation, we get that m ∠ YXZ=5x-12 as well. Finally, we can substitute x=12 into this equation to find the measure of the angle.
