Question

fill in the blank 1 point
4.
$m\angle w = $
$m\angle x = $
$m\angle z = $
$m\angle w = $ type your answer...
$m\angle x = $ type your answer...
$m\angle z = $ type your answer...

Explanation:

Step1: Find ∠XYZ's interior angle

The exterior angle at Y is 146°, so the interior angle is supplementary:
$m\angle XYZ = 180^\circ - 146^\circ = 34^\circ$

Step2: Identify isosceles triangle property

Sides $WZ$ and $XY$ are congruent, sides $WX$ and $ZY$ are congruent, so $WXYZ$ is an isosceles trapezoid. Thus, $m\angle W = m\angle X$, and $m\angle X = m\angle Z$? No, correct: in isosceles trapezoid, base angles are equal, so $\angle W = \angle X$, $\angle XYZ = \angle Z$. Wait, no: the parallel sides are $WX$ and $ZY$ (arrow marks), so base angles: $\angle W$ and $\angle Z$ are one pair, $\angle X$ and $\angle XYZ$ are the other. Wait, no, the congruent sides are $WZ$ and $XY$, so legs are $WZ, XY$, bases are $WX, ZY$. So base angles: $\angle W \cong \angle Z$, $\angle X \cong \angle XYZ$. Wait, no, let's use sum of interior angles of quadrilateral is $360^\circ$. Let $m\angle W = m\angle Z = x$, $m\angle X = m\angle XYZ = 34^\circ$. Wait no, the exterior angle is at Y, so interior $\angle Y = 34^\circ$, and since $WX \parallel ZY$, $\angle X + \angle Y = 180^\circ$, so $\angle X = 180 - 34 = 146^\circ$. Then since it's isosceles trapezoid, $\angle W = \angle Z$, sum of angles: $146 + 34 + x + x = 360$

Step3: Calculate ∠W and ∠Z

$180 + 2x = 360$
$2x = 360 - 180 = 180$
$x = \frac{180}{2} = 90$? No, wait no: $146 + 34 = 180$, so $2x = 360 - 180 = 180$, $x=90$? No, wait, no: if $WX \parallel ZY$, then $\angle W + \angle Z = 180$? No, no, consecutive angles between parallel lines are supplementary. So $\angle W + \angle ZY W = 180$, but $\angle ZYW$ is $\angle Y$? No, the sides: $WX$ and $ZY$ are parallel (arrow marks), so $\angle W$ and $\angle Z$ are same-side interior angles? No, vertices are W, X, Y, Z. So the quadrilateral is W-X-Y-Z-W. So sides WX, XY, YZ, ZW. WX and YZ are parallel (arrows), XY and ZW are congruent (tick marks). So it's an isosceles trapezoid with bases WX and YZ, legs XY and ZW. So base angles: $\angle W \cong \angle X$, $\angle Y \cong \angle Z$. Then exterior angle at Y is 146°, so interior $\angle Y = 180 - 146 = 34^\circ$, so $\angle Z = 34^\circ$. Then sum of angles: $\angle W + \angle X + 34 + 34 = 360$, and $\angle W = \angle X$, so $2\angle W = 360 - 68 = 292$, $\angle W = \frac{292}{2} = 146^\circ$, $\angle X = 146^\circ$. Yes, that makes sense, because parallel sides WX and YZ, so $\angle W + \angle Z = 146 + 34 = 180$, which is supplementary, correct.

Answer:

$m\angle W = 146^\circ$
$m\angle X = 146^\circ$
$m\angle Z = 34^\circ$