Question

quadrilateral wxyz is an isosceles trapezoid and ( mangle w = 5w - 66^circ ). what is the value of ( w )?

Explanation:

Step1: Recall property of isosceles trapezoid

In an isosceles trapezoid, consecutive angles between the bases are supplementary (sum to \(180^\circ\)). Here, \(\angle W\) and \(\angle Z\) are consecutive angles, so \(m\angle W + m\angle Z = 180^\circ\). We know \(m\angle Z = 144^\circ\) and \(m\angle W = 5w - 66^\circ\).

Step2: Set up the equation

Substitute the known values into the supplementary angle equation: \(5w - 66 + 144 = 180\).

Step3: Simplify and solve for \(w\)

First, simplify the left side: \(5w + 78 = 180\). Then, subtract 78 from both sides: \(5w = 180 - 78 = 102\). Finally, divide both sides by 5: \(w=\frac{102}{5}= 42\)? Wait, no, wait. Wait, 180 - 144 is 36? Wait, no, I made a mistake. Wait, \(\angle W\) and \(\angle Z\) are supplementary, so \(m\angle W = 180 - 144 = 36^\circ\)? Wait, no, wait the angle given is \(\angle Z = 144^\circ\), so \(\angle W\) should be \(180 - 144 = 36^\circ\)? Wait, no, the equation is \(5w - 66 = 180 - 144\). Let's correct that.

Step1 (corrected): Recall supplementary angles

In isosceles trapezoid, adjacent angles on the same side are supplementary. So \(m\angle W + m\angle Z = 180^\circ\). Given \(m\angle Z = 144^\circ\), so \(m\angle W = 180 - 144 = 36^\circ\)? Wait, no, the expression for \(m\angle W\) is \(5w - 66\), so set \(5w - 66 = 180 - 144\).

Step2 (corrected): Solve the equation

Calculate \(180 - 144 = 36\), so the equation is \(5w - 66 = 36\). Add 66 to both sides: \(5w = 36 + 66 = 102\). Then divide by 5: \(w=\frac{102}{5}\)? Wait, no, that can't be. Wait, maybe I mixed up the angles. Wait, maybe \(\angle W\) and \(\angle X\) or something else? Wait, no, the diagram: W and Z are the top base? Wait, the trapezoid is WXYZ, so the sides are WX, XY, YZ, ZW. So the bases are WZ and XY? So angles at W and Z are on the top base, and angles at X and Y on the bottom? Wait, no, in a trapezoid, the bases are the two parallel sides. So if WXYZ is isosceles, then WZ and XY are the bases (parallel), so angles at W and X are adjacent, angles at Z and Y are adjacent. Wait, maybe I had the angles wrong. Wait, the given angle is at Z, 144 degrees. So angle Z and angle W: are they adjacent? If WZ and XY are parallel, then angle W and angle X are supplementary, angle Z and angle Y are supplementary. Wait, maybe the diagram shows that angle Z is 144, so angle W is equal to angle X? No, in isosceles trapezoid, base angles are equal. So angles at W and Z are equal? No, no: in isosceles trapezoid, the base angles (angles adjacent to each base) are equal. So if the two bases are WZ and XY, then angles at W and Z are on the top base, angles at X and Y on the bottom. So angle W and angle X are supplementary, angle Z and angle Y are supplementary, and angle W = angle Z? No, that's not right. Wait, no, in an isosceles trapezoid, each pair of base angles is equal. So if the legs are WX and YZ, then the bases are WZ and XY. So angle at W and angle at Z are adjacent to the top base, angle at X and angle at Y adjacent to the bottom base. Then angle W = angle Z? No, that's not supplementary. Wait, I think I made a mistake earlier. Let's check the property again: In an isosceles trapezoid, consecutive angles between the bases are supplementary. So if the bases are WZ and XY (parallel), then angle W (on top base) and angle X (on bottom base) are supplementary, angle Z (on top base) and angle Y (on bottom base) are supplementary. And angle W = angle Z, angle X = angle Y. Wait, no, that's not. Wait, no: the base angles are equal. So angle at W and angle at Z: are they base angles? No, the base angles are the angles adjacent to each base. So if the two bases are WZ (top) and XY (bottom), then the legs are WX and YZ. So the base angles are angle W and angle X (adjacent to base WZ and XY), and angle Z and angle Y (adjacent to base WZ and XY). Wait, no, maybe the diagram is W connected to X, X to Y, Y to Z, Z to W. So the sides are WX, XY, YZ, ZW. So the parallel sides are WZ and XY (so WZ || XY). Then, angle at W (between WZ and WX) and angle at X (between XY and WX) are supplementary. Angle at Z (between WZ and ZY) and angle at Y (between XY and ZY) are supplementary. And since it's isosceles, angle at W = angle at Z, angle at X = angle at Y. Wait, that makes sense. So angle W = angle Z? No, that would mean they are equal, not supplementary. Wait, I'm confused. Let's look at the given angle: angle Z is 144 degrees. So if angle W and angle Z are equal (since it's isosceles, base angles equal), but that would mean 144, but then supplementary would be 36. Wait, no, maybe the parallel sides are WX and YZ. Oh! Maybe I got the bases wrong. Let's re-examine the diagram: the trapezoid is labeled W, X, Y, Z. So the vertices are W, X, Y, Z in order. So the sides are WX, XY, YZ, ZW. So if it's an isosceles trapezoid, the non-parallel sides are WX and YZ (the legs), and the parallel sides are XY and WZ (the bases). So then, angles at W and Z are adjacent to the base WZ, and angles at X and Y are adjacent to the base XY. In an isosceles trapezoid, angles adjacent to each leg are supplementary. So angle W (adjacent to leg WX) and angle X (adjacent to leg WX) are supplementary. Angle Z (adjacent to leg YZ) and angle Y (adjacent to leg YZ) are supplementary. Also, angle W = angle Z, angle X = angle Y. Wait, now I'm really confused. Let's check the problem again: it says "Quadrilateral WXYZ is an isosceles trapezoid and \(m\angle W = 5w - 66^\circ\). What is the value of w?" The diagram shows angle at Z is 144 degrees. So maybe angle W and angle Z are supplementary. So \(m\angle W + m\angle Z = 180^\circ\). So \(5w - 66 + 144 = 180\). Let's solve that: \(5w + 78 = 180\), \(5w = 180 - 78 = 102\), \(w = 102 / 5 = 20.4\)? No, that can't be. Wait, maybe angle W is equal to angle Z? But 144, then \(5w - 66 = 144\), so \(5w = 210\), \(w = 42\). Ah! That must be it. I was wrong earlier. In an isosceles trapezoid, the base angles (angles on each base) are equal. So if WZ and XY are the bases, then angles at W and Z are on the base WZ, so they are equal. Wait, no, base angles are on each base. So if the two bases are WZ (top) and XY (bottom), then the base angles are angle W and angle X (on base WZ and XY, adjacent to leg WX), and angle Z and angle Y (on base WZ and XY, adjacent to leg YZ). Wait, no, I think the key is that in the diagram, angle Z is 144, and angle W is given as \(5w - 66\). If it's an isosceles trapezoid, maybe angle W and angle Z are equal? But that would mean 144, but then the equation would be \(5w - 66 = 144\), so \(5w = 210\), \(w = 42\). Let's check: \(5*42 - 66 = 210 - 66 = 144\), which is equal to angle Z. So that makes sense. So my initial mistake was thinking they are supplementary, but actually, if the bases are WX and YZ, then WZ and XY are the legs? No, no, the labels are W, X, Y, Z. So the correct property: in an isosceles trapezoid, the base angles are equal. So if angle Z is 144, then angle W is also 144? But that would mean the other angles are 36, which are equal. Wait, but the sum of interior angles in a quadrilateral is 360. So 144 + 144 + 36 + 36 = 360, which works. So that must be it. So angle W = angle Z = 144 degrees. So set \(5w - 66 = 144\).

Step1: Set angle W equal to angle Z (isosceles trapezoid base angles equal)

Since WXYZ is an isosceles trapezoid, \(m\angle W = m\angle Z\). Given \(m\angle Z = 144^\circ\) and \(m\angle W = 5w - 66^\circ\), we have \(5w - 66 = 144\).

Step2: Solve for \(w\)

Add 66 to both sides: \(5w = 144 + 66 = 210\). Divide both sides by 5: \(w = \frac{210}{5} = 42\).

Answer:

\(w = \boxed{42}\)