Question

properties of rotations (and all other rigid motions)

1. rotations transform lines into lines, segments into segments, and rays into rays.
2. rotations preserve the distance between points and hence the length of line segments.
3. rotations preserve angles between lines, rays, and segments.

when we put these three properties together, it implies that rotations will preserve size (property #2 above) and preserve shape (property #3 above). rotations by 180° have very important properties. recall that two distinct lines either intersect once or not at all (in which case they are called parallel).
exercise #3: given $overline{ab}$ shown below. construct $overline{cd}$ such that $r_{e,180^{circ}}(a)\to c$ and $r_{e,180^{circ}}(b)\to d$. use only a compass and a straightedge and leave construction marks.
what appears to be true about $overline{ab}$ and $overline{cd}$?
geometric fact: a line which is rotated 180° about a point not on the line will always result in another line that is parallel to the original. a line which is rotated 180° about a point that is on the line will simply produce the same line.
we can use facts about rotations to prove one of the more important properties of parallel lines. we will work through both the thinking and the property in the next exercise.
exercise #4: recall that when two parallel lines are crossed by third line (called a transversal) it forms alternate interior angle pairs. in the diagram below, parallel lines $overline{ab}$ and $overline{cd}$ are crossed by line $m$. the mid - point of $overline{bc}$ is point $e$. angles $angle abe$ and $angle dce$ are such a pair.
(a) if we rotated $overline{ec}$ around $e$ by 180°, where would it land?
(b) if we rotated $overline{cd}$ about $e$ by 180°, would it coincide (lie on top of) $overline{ab}$? why or why not?
given your two answers, what must be true about the angle pairs $angle abe$ and $angle dce$? why?

Explanation:

Step1: Analyze Exercise #3

Since rotation by 180 - degree about a point not on a line results in a parallel line, $\overline{AB}$ and $\overline{CD}$ are parallel.

Step2: Analyze Exercise #4 (a)

Point $E$ is the mid - point of $\overline{BC}$. When $\overline{EC}$ is rotated 180 degrees around $E$, it will land on $\overline{EB}$ because of the property of 180 - degree rotation about the mid - point of a line segment.

Step3: Analyze Exercise #4 (b)

Since $\overline{AB}$ and $\overline{CD}$ are parallel and rotation by 180 degrees about the mid - point $E$ of $\overline{BC}$ preserves orientation and parallelism, $\overline{CD}$ rotated 180 degrees about $E$ will coincide with $\overline{AB}$. Also, because of the properties of parallel lines and 180 - degree rotation, $\angle ABE$ and $\angle DCE$ are equal. This is because 180 - degree rotation preserves angles and the two angles are alternate interior angles of parallel lines $\overline{AB}$ and $\overline{CD}$ cut by transversal $m$.

Answer:

For Exercise #3: $\overline{AB}$ and $\overline{CD}$ are parallel.
For Exercise #4 (a): $\overline{EC}$ rotated 180 degrees around $E$ will land on $\overline{EB}$.
For Exercise #4 (b): $\overline{CD}$ rotated 180 degrees about $E$ will coincide with $\overline{AB}$ because $\overline{AB}\parallel\overline{CD}$ and 180 - degree rotation about the mid - point of the segment connecting them preserves parallelism and orientation. Also, $\angle ABE=\angle DCE$ due to properties of parallel lines and 180 - degree rotation.