Question

find the measures of $\angle x$ and $\angle y$.
$m\angle x= \boldsymbol{47} \times$ $^\circ$
$m\angle y= \boldsymbol{47} \times$ $^\circ$

Explanation:

Step1: Recall the sum of interior angles of a pentagon? No, this is a pentagon? Wait, the figure has five sides? Wait, no, looking at the vertices: X, Z, V, W, Y. So it's a pentagon? Wait, no, X to Z to V to W to Y to X. So five sides. The sum of interior angles of a polygon is $(n - 2)\times180^\circ$, where $n$ is the number of sides. For $n = 5$, sum is $(5 - 2)\times180^\circ = 540^\circ$.

Step2: Identify known angles. $\angle Z = 164^\circ$, $\angle V = 102^\circ$, $\angle W = 90^\circ$ (right angle), $\angle X$ and $\angle Y$ are what we need to find. Wait, but looking at the figure, $\angle X$ and $\angle Y$: wait, maybe it's a pentagon? Wait, no, maybe I miscounted. Wait, X, Z, V, W, Y: five vertices, so pentagon. So sum of interior angles is $540^\circ$.

Wait, but in the figure, $\angle W$ is a right angle (90°), $\angle Z = 164°$, $\angle V = 102°$, and $\angle X$ and $\angle Y$: wait, maybe $\angle X$ and $\angle Y$ are equal? Wait, no, let's check again. Wait, maybe it's a pentagon with two right angles? Wait, no, the figure shows $\angle W$ as right angle, $\angle X$ and $\angle Y$: maybe $\angle X$ and $\angle Y$ are right angles? No, the initial answer was wrong. Wait, let's recalculate.

Sum of interior angles for a pentagon: $(5 - 2)\times180 = 540^\circ$.

Known angles: $\angle Z = 164^\circ$, $\angle V = 102^\circ$, $\angle W = 90^\circ$, $\angle X$ and $\angle Y$: let's denote $\angle X = x$, $\angle Y = y$. Wait, but maybe $\angle X$ and $\angle Y$ are equal? Wait, no, maybe the figure is a pentagon with $\angle X$ and $\angle Y$: wait, maybe I made a mistake. Wait, no, looking at the figure again: X to Z to V to W to Y to X. So five sides. So sum is 540.

So $x + y + 164 + 102 + 90 = 540$.

Simplify: $x + y + 356 = 540$. So $x + y = 540 - 356 = 184$. Wait, but that can't be. Wait, maybe it's a quadrilateral? Wait, no, the vertices are X, Z, V, W, Y: five points. Wait, maybe the figure is a pentagon, but maybe $\angle X$ and $\angle Y$ are both right angles? No, that doesn't fit. Wait, maybe I miscounted the number of sides. Wait, X to Z to V to W to Y to X: that's five sides. Wait, maybe the figure is a pentagon with $\angle X$ and $\angle Y$: let's check again.

Wait, maybe the figure is a pentagon, but the user's initial answer was 47, which is wrong. Wait, let's recalculate.

Wait, maybe it's a pentagon with two right angles? No, $\angle W$ is 90, maybe $\angle X$ and $\angle Y$: wait, maybe the figure is a pentagon, but I made a mistake. Wait, let's check the sum again.

Wait, $(5 - 2)\times180 = 540$. So 540.

Known angles: 164 (Z), 102 (V), 90 (W), and two angles X and Y. Wait, maybe X and Y are equal? Then $2x + 164 + 102 + 90 = 540$. So $2x + 356 = 540$. Then $2x = 540 - 356 = 184$. So $x = 92$. Wait, that makes sense. Wait, maybe the figure has $\angle X$ and $\angle Y$ equal. So let's check:

164 + 102 + 90 + 92 + 92 = 164 + 102 = 266; 266 + 90 = 356; 356 + 92 = 448; 448 + 92 = 540. Yes, that works. So maybe the initial mistake was thinking it's a different polygon. So the correct approach is:

Sum of interior angles of a pentagon: $(5 - 2)\times180 = 540^\circ$.

Known angles: $\angle Z = 164^\circ$, $\angle V = 102^\circ$, $\angle W = 90^\circ$, $\angle X$ and $\angle Y$ are equal (maybe the figure is symmetric). So let $x = \angle X$, $y = \angle Y$, and $x = y$.

So $x + y + 164 + 102 + 90 = 540$.

Since $x = y$, substitute: $2x + 356 = 540$.

Subtract 356: $2x = 540 - 356 = 184$.

Divide by 2: $x = 92$. So $\angle X = 92^\circ$, $\angle Y = 92^\circ$.

Wait, that makes sense. So the initial answer of 47 was wrong. Let's verify:

164 + 102 + 90 + 92 + 92 = 164 + 102 = 266; 266 + 90 = 356; 356 + 92 = 448; 448 + 92 = 540. Correct.

So the steps are:

1. Calculate the sum of interior angles of a pentagon: $(5 - 2)\times180 = 540^\circ$.

2. Sum the known angles: $164 + 102 + 90 = 356^\circ$.

3. Subtract from the total to find the sum of $\angle X$ and $\angle Y$: $540 - 356 = 184^\circ$.

4. Since $\angle X$ and $\angle Y$ appear to be equal (from the figure's symmetry, both having right-angle-like marks? Wait, no, the initial marks: $\angle X$ and $\angle Y$ have right-angle marks? Wait, looking at the figure, $\angle X$ and $\angle Y$: the original figure shows $\angle X$ and $\angle Y$ with right-angle marks? Wait, no, the user's figure: X has a right-angle mark? Wait, maybe I misread. Wait, the figure: X, Z, V, W, Y. $\angle X$: the corner at X has a right-angle mark? $\angle Y$: corner at Y has a right-angle mark? $\angle W$: corner at W has a right-angle mark. Wait, that would be three right angles. Then it's a pentagon with three right angles, $\angle Z = 164$, $\angle V = 102$. Wait, no, three right angles (90 each), so 3*90 = 270, plus 164 + 102 = 266, total 270 + 266 = 536, which is less than 540. So that can't be. Wait, maybe the figure is a quadrilateral? Wait, no, five vertices. Wait, maybe the figure is a pentagon, but the right-angle marks are at X, Y, W. So three right angles: 90, 90, 90. Then $\angle Z = 164$, $\angle V = 102$. Then sum of angles: 90 + 90 + 90 + 164 + 102 = 90*3 = 270; 270 + 164 = 434; 434 + 102 = 536. Then the fifth angle? Wait, no, five vertices, so five angles. Wait, I'm confused. Wait, let's count the sides again: X to Z (1), Z to V (2), V to W (3), W to Y (4), Y to X (5). So five sides, pentagon. So five angles: $\angle X$, $\angle Z$, $\angle V$, $\angle W$, $\angle Y$. So five angles. So sum is 540. So if $\angle X$, $\angle Y$, $\angle W$ are right angles (90 each), then 3*90 = 270, plus 164 + 102 = 266, total 270 + 266 = 536, which is 4 less than 540. So that's not possible. Therefore, my initial assumption about the right-angle marks is wrong.

Wait, looking at the user's figure: $\angle W$ has a right-angle mark, $\angle X$ and $\angle Y$: maybe $\angle X$ and $\angle Y$ are not right angles. Wait, the user's initial answer was 47, which is wrong. Let's recalculate correctly.

Sum of interior angles of a pentagon: $(5 - 2)\times180 = 540^\circ$.

Known angles: $\angle Z = 164^\circ$, $\angle V = 102^\circ$, $\angle W = 90^\circ$ (right angle). Let $\angle X = x$, $\angle Y = y$. Then:

$x + y + 164 + 102 + 90 = 540$

$x + y + 356 = 540$

$x + y = 540 - 356 = 184$

Now, if the figure is symmetric, maybe $x = y$, so $x = y = 92$. That makes sense. So the correct measures are $\angle X = 92^\circ$ and $\angle Y = 92^\circ$.

Answer:

$m\angle X = \boxed{92}^\circ$, $m\angle Y = \boxed{92}^\circ$