1. **Problem statement:** Find the angle between the line $MA$ and the plane $ABCD$ in a cube $ABCDEFGH$, where $M$ is the midpoint of edge $GH$. 2. **Understanding the problem:** - The cube has vertices $A, B, C, D, E, F, G, H$. - $M$ is the midpoint of edge $GH$. - We want the angle between line $MA$ and plane $ABCD$. 3. **Key idea:** The angle between a line and a plane is the complement of the angle between the line and the plane's normal vector. 4. **Step 1: Assign coordinates** Assume the cube has side length 1 for simplicity. - Let $A = (0,0,0)$ - $B = (1,0,0)$ - $C = (1,1,0)$ - $D = (0,1,0)$ - $E = (0,0,1)$ - $F = (1,0,1)$ - $G = (1,1,1)$ - $H = (0,1,1)$ 5. **Step 2: Find point $M$** $M$ is midpoint of $G(1,1,1)$ and $H(0,1,1)$: $$M = \left(\frac{1+0}{2}, \frac{1+1}{2}, \frac{1+1}{2}\right) = (0.5, 1, 1)$$ 6. **Step 3: Vector $\overrightarrow{MA}$** $$\overrightarrow{MA} = A - M = (0 - 0.5, 0 - 1, 0 - 1) = (-0.5, -1, -1)$$ 7. **Step 4: Normal vector to plane $ABCD$** Plane $ABCD$ lies in the $xy$-plane at $z=0$, so its normal vector is: $$\vec{n} = (0,0,1)$$ 8. **Step 5: Find angle $\theta$ between $\overrightarrow{MA}$ and $\vec{n}$** Use the dot product formula: $$\cos \theta = \frac{\overrightarrow{MA} \cdot \vec{n}}{|\overrightarrow{MA}| |\vec{n}|}$$ Calculate dot product: $$\overrightarrow{MA} \cdot \vec{n} = (-0.5)(0) + (-1)(0) + (-1)(1) = -1$$ Calculate magnitudes: $$|\overrightarrow{MA}| = \sqrt{(-0.5)^2 + (-1)^2 + (-1)^2} = \sqrt{0.25 + 1 + 1} = \sqrt{2.25} = 1.5$$ $$|\vec{n}| = 1$$ So: $$\cos \theta = \frac{-1}{1.5 \times 1} = -\frac{2}{3}$$ 9. **Step 6: Find $\theta$** $$\theta = \cos^{-1} \left(-\frac{2}{3}\right) \approx 131.8^\circ$$ 10. **Step 7: Find angle between line and plane** The angle between the line and the plane is the complement of the angle between the line and the normal vector: $$\alpha = 90^\circ - (\theta - 90^\circ) = 180^\circ - \theta = 180^\circ - 131.8^\circ = 48.2^\circ$$ **Final answer:** $$\boxed{48.2^\circ}$$ This is the angle between line $MA$ and plane $ABCD$, correct to 1 decimal place.