Homework #5

Hard copy due in class Wednesday, April 2

See homework guidelines and tips here

Hint for Problems 1-4: you should rely on the definitions of the relevant objects for each problem as given in class.

1. Consider a configuration of two lines in 3-space. We have seen two situations where a plane can pass through these two lines. Describe each of these two possible situations (what must the two lines satisfy in order for a plane to pass through them). Be precise and include all details. Then describe a third situation where two lines are configured in such a way that no plane passes through them.
2. Let $P_1$ and $P_2$ be two planes in 3-space that are parallel. Assume that $Q$ is a third plane that intersects $P_1$ in ${\ell }_1$ and $P_2$ in ${\ell }_2$. Prove or disprove: ${\ell }_1$ and ${\ell }_2$ are parallel in 3-space.
3. Assume that a plane $P$ and a line ${\ell }_1$ are both perpendicular to another line ${\ell }_2$. Assume also that ${\ell }_1$ does not lie in $P$. Prove that $P$ and ${\ell }_1$ are parallel to one another.
4. Consider two dihedral angles that are vertically opposite to one another. Prove that these dihedral angles are congruent to one another.
5. OpenLab mini-assignment: typing in $LATEX$ (instructions here)