Prove that an equilateral triangle can be constructed on any given line segment by using Euclid's $3^{\mathrm{rd}}$ postulate.

In the figure, $\mathrm{A}\mathrm{B}=\mathrm{B}\mathrm{C}, \mathrm{P}$ is midpoint of $\mathrm{A}\mathrm{B}$ and $\mathrm{Q}$ is midpoint of $\mathrm{B}\mathrm{C}.$ Show that $\mathrm{A}\mathrm{P}=\mathrm{Q}\mathrm{C}$.

In the adjoining figure, name :

Two pairs of intersecting lines and their corresponding points of Intersection.

In the following figure, if $\mathrm{A}\mathrm{C}=\mathrm{B}\mathrm{D},$ then prove that $\mathrm{A}\mathrm{B}=\mathrm{C}\mathrm{D}.$

Does Euclid fifth postulate imply the existence of parallel lines? Explain.

Boundaries of solids are