find the smallest angle of $\triangle wxy$. assume that $c$ is a positive number.

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# Explanation: ## Step1: Recall the triangle side-angle relationship In a triangle, the smallest angle is opposite the shortest side. So we first compare the lengths of the sides: $44c$, $57c$, and $60c$. Since $c>0$, we can compare the coefficients. $44 < 57 < 60$, so the shortest side is $44c$, which is opposite angle $X$? Wait, no, let's label the triangle. The sides: $WY = 44c$, $XY = 57c$, $WX = 60c$. So side $WY = 44c$ is opposite angle $X$? Wait, no, in triangle $WXY$, vertex $W$, $X$, $Y$. Side opposite $W$ is $XY = 57c$, side opposite $X$ is $WY = 44c$, side opposite $Y$ is $WX = 60c$. So the shortest side is $WY = 44c$, opposite angle $X$? Wait, no, wait: side opposite angle $W$ is $XY$ (length $57c$), side opposite angle $X$ is $WY$ (length $44c$), side opposite angle $Y$ is $WX$ (length $60c$). So the shortest side is $44c$ (opposite angle $X$)? Wait, no, wait, maybe I mixed up. Wait, the side $WY$ is between $W$ and $Y$, length $44c$. Side $XY$ is between $X$ and $Y$, length $57c$. Side $WX$ is between $W$ and $X$, length $60c$. So angle at $X$ is between $WX$ and $XY$, so the side opposite angle $X$ is $WY$ (length $44c$). Angle at $W$ is between $WY$ and $WX$, so side opposite angle $W$ is $XY$ (length $57c$). Angle at $Y$ is between $WY$ and $XY$, so side opposite angle $Y$ is $WX$ (length $60c$). So the shortest side is $44c$ (opposite angle $X$), so angle $X$ is the smallest? Wait, no, wait: in a triangle, the larger the side, the larger the angle opposite it. So the shortest side is $44c$, so the angle opposite to it (angle $X$) is the smallest? Wait, no, wait: side $WY = 44c$ is opposite angle $X$. Side $XY = 57c$ is opposite angle $W$. Side $WX = 60c$ is opposite angle $Y$. So comparing the sides: $44c < 57c < 60c$, so the angles opposite them: angle $X$ (opposite $44c$) < angle $W$ (opposite $57c$) < angle $Y$ (opposite $60c$). Wait, but let's confirm. So the smallest angle is opposite the shortest side. So shortest side is $44c$, opposite angle $X$? Wait, no, wait, maybe I got the labels wrong. Let me re-express: Vertices: $W$, $X$, $Y$. Sides: - $W$ to $Y$: $WY = 44c$ - $Y$ to $X$: $YX = 57c$ - $X$ to $W$: $XW = 60c$ Angles: - Angle at $W$ (∠$W$): between $WY$ and $XW$, so opposite side is $YX = 57c$ - Angle at $X$ (∠$X$): between $XW$ and $YX$, so opposite side is $WY = 44c$ - Angle at $Y$ (∠$Y$): between $WY$ and $YX$, so opposite side is $XW = 60c$ So since $44c < 57c < 60c$, the angles opposite them are ∠$X$ < ∠$W$ < ∠$Y$. Wait, but that would mean the smallest angle is ∠$X$? Wait, no, wait, maybe I made a mistake. Wait, the side opposite angle $W$ is $YX$ (57c), side opposite angle $X$ is $WY$ (44c), side opposite angle $Y$ is $XW$ (60c). So yes, the smallest side is $44c$ (opposite angle $X$), so angle $X$ is the smallest? Wait, but let's check the side lengths again. Wait, the problem is to find the smallest angle. So according to the triangle side-angle relationship: in a triangle, the larger the length of a side, the larger the measure of the angle opposite that side. So we need to find which side is the shortest, then the angle opposite that side is the smallest. So the sides are $44c$, $57c$, $60c$. Since $c > 0$, we can compare the coefficients: $44 < 57 < 60$, so $44c < 57c < 60c$. Therefore, the shortest side is $44c$, which is $WY$. The angle opposite $WY$ is angle $X$ (because $WY$ is between $W$ and $Y$, so the angle at $X$ is opposite $WY$). Therefore, angle $X$ is the smallest angle? Wait, no, wait, maybe I mixed up the sides. Wait, let's look at the diagram: the side labeled $44c$ is $WY$, $57c$ is $XY$, $60c$ is $WX$. So angle at $X$ is between $WX$ (60c) and $XY$ (57c), so the side opposite angle $X$ is $WY$ (44c). Angle at $W$ is between $WY$ (44c) and $WX$ (60c), so side opposite angle $W$ is $XY$ (57c). Angle at $Y$ is between $WY$ (44c) and $XY$ (57c), so side opposite angle $Y$ is $WX$ (60c). So yes, the shortest side is $44c$ (opposite angle $X$), so angle $X$ is the smallest? Wait, but let's confirm with the side lengths. Wait, maybe I got the angle labels wrong. Wait, the triangle is $WXY$, so the vertices are $W$, $X$, $Y$. So the sides: $WX$, $XY$, $YW$. So $WX = 60c$, $XY = 57c$, $YW = 44c$. Then angle at $W$ is between $YW$ and $WX$, so opposite side $XY$ (57c). Angle at $X$ is between $WX$ and $XY$, opposite side $YW$ (44c). Angle at $Y$ is between $YW$ and $XY$, opposite side $WX$ (60c). So since $YW = 44c$ is the shortest side, angle at $X$ (opposite $YW$) is the smallest angle. Wait, but the problem is to find the smallest angle. So the answer should be angle $X$? Wait, but let's check again. Wait, maybe the side $44c$ is $WY$, $57c$ is $XY$, $60c$ is $WX$. So the angles: angle at $W$ is between $WY$ and $WX$, so sides $WY = 44c$, $WX = 60c$, so the angle at $W$ is between these two sides, so the side opposite angle $W$ is $XY = 57c$. Angle at $X$ is between $WX = 60c$ and $XY = 57c$, so side opposite angle $X$ is $WY = 44c$. Angle at $Y$ is between $WY = 44c$ and $XY = 57c$, so side opposite angle $Y$ is $WX = 60c$. So since $44c < 57c < 60c$, the angles opposite them are angle $X$ (opposite $44c$) < angle $W$ (opposite $57c$) < angle $Y$ (opposite $60c$). Therefore, the smallest angle is angle $X$. Wait, but let's make sure. So the key is: in a triangle, the smallest angle is opposite the shortest side. So we find the shortest side, then the angle opposite to it is the smallest. So the sides are $44c$, $57c$, $60c$. Shortest is $44c$, opposite angle $X$, so angle $X$ is the smallest. # Answer: The smallest angle is $\angle X$ (opposite the shortest side $44c$).

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Step1: Recall the triangle side-angle relationship

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