1. **Stating the problem:** Construct a quadrilateral $WXYZ$ with given side lengths $WX=5.3$, $XY=4.6$, $WZ=3.9$, and angle $\angle X = 75^\circ$. 2. **Understanding the problem:** We have three sides and one angle. We need to find the positions of points $W$, $X$, $Y$, and $Z$ to form the quadrilateral. 3. **Step-by-step construction:** - Place point $W$ at the origin $(0,0)$. - Draw segment $WX$ along the positive x-axis with length $5.3$, so $X$ is at $(5.3,0)$. - At point $X$, construct an angle of $75^\circ$ with the segment $WX$. - From $X$, draw segment $XY$ of length $4.6$ at $75^\circ$ to $WX$. Calculate coordinates of $Y$: $$Y_x = 5.3 + 4.6 \cos 75^\circ$$ $$Y_y = 0 + 4.6 \sin 75^\circ$$ Using $\cos 75^\circ \approx 0.2588$ and $\sin 75^\circ \approx 0.9659$: $$Y_x \approx 5.3 + 4.6 \times 0.2588 = 5.3 + 1.19 = 6.49$$ $$Y_y \approx 0 + 4.6 \times 0.9659 = 4.44$$ - Point $Y$ is approximately at $(6.49, 4.44)$. 4. **Locate point $Z$:** We know $WZ=3.9$. Point $Z$ must satisfy distance $WZ=3.9$ and form a quadrilateral with points $W$, $X$, and $Y$. 5. **Summary:** The quadrilateral $WXYZ$ has vertices: $$W = (0,0), X = (5.3,0), Y = (6.49,4.44), Z = ?$$ Point $Z$ lies on the circle centered at $W$ with radius $3.9$. 6. **Final note:** Without additional information (like length $YZ$ or angle at $W$ or $Z$), the exact position of $Z$ cannot be uniquely determined. Thus, the construction is partially complete with $W$, $X$, and $Y$ located, and $Z$ constrained to a circle of radius $3.9$ centered at $W$.