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.
