1. **Problem Statement:** (i) Show that the combination of reflections in two parallel lines, distance $\frac{d}{2}$ apart, results in a translation through distance $d$ perpendicular to the lines. (ii) Show that the combination of reflections in two lines meeting at angle $\frac{\theta}{2}$ results in a rotation through angle $\theta$ about their intersection. 2. **Reflection in Parallel Lines (Part i):** - When a point is reflected across the first line, it moves to a symmetric position on the opposite side. - Reflecting this image across the second parallel line (distance $\frac{d}{2}$ away) moves the point again by $\frac{d}{2}$ in the same direction. - Total movement is $\frac{d}{2} + \frac{d}{2} = d$. - This combined effect is a translation by distance $d$ perpendicular to the lines. 3. **Reflection in Intersecting Lines (Part ii):** - Let the two lines intersect at point $O$ with angle $\frac{\theta}{2}$ between them. - Reflecting a point across the first line changes its position symmetrically. - Reflecting the image across the second line rotates the point about $O$ by twice the angle between the lines. - Since the angle between lines is $\frac{\theta}{2}$, the total rotation is $2 \times \frac{\theta}{2} = \theta$. - Thus, the combination of reflections is a rotation through angle $\theta$ about $O$. 4. **Summary:** - Two reflections in parallel lines distance $\frac{d}{2}$ apart equal a translation by $d$ perpendicular to the lines. - Two reflections in lines intersecting at $\frac{\theta}{2}$ equal a rotation by $\theta$ about their intersection. Final answers: (i) Translation distance $d$ perpendicular to lines. (ii) Rotation angle $\theta$ about intersection point.