1. **State the problem:** We need to find the size of angle $\theta$ in a triangle with two sides measuring 39.5 cm and 45.3 cm, where the angle $\theta$ is between these two sides and the triangle includes a right angle. 2. **Identify what is given and what to find:** Given sides $a = 39.5$ cm and $b = 45.3$ cm, angle $\theta$ is opposite to the side across from the right angle. 3. **Apply the cosine rule:** Since we have two sides and the included angle is $\theta$, the cosine rule states: $$c^2 = a^2 + b^2 - 2ab\cos\theta$$ However, the triangle contains a right angle, so the side opposite the right angle is the hypotenuse $c = 45.3$ cm or $39.5$ cm, but the problem states the right angle is at the vertex adjacent to $\theta$. 4. **Use trigonometry for right triangles:** Since one angle is 90°, the other two angles sum to 90°. The given sides adjacent to $\theta$ are 39.5 cm and 45.3 cm. 5. **Calculate $\theta$ using tangent:** Opposite side = 39.5 cm, adjacent side = 45.3 cm for angle $\theta$. $$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{39.5}{45.3}$$ 6. **Find $\theta$ by inverse tangent:** $$\theta = \tan^{-1}\left(\frac{39.5}{45.3}\right)$$ 7. **Calculate the value:** $$\frac{39.5}{45.3} \approx 0.872$$ $$\theta = \tan^{-1}(0.872) \approx 41.3^\circ$$ **Final answer:** $$\boxed{41.3^\circ}$$