This reasoning implies that the rotation maps $m$ to $\ell$ and $\ell$ to $m$.
The parallel postulate says that there is only one line through $Q$ (respectively $P$) parallel to $k$ and this is $\ell$ (respectively $m$). The 180 degree rotations of $m$ and $\ell$ are parallel to $k$ and pass through $Q$ and $P$ respectively.
This 180 degree rotation maps $k$ to itself and so the images of $m$ and $\ell$ are parallel to $k$. Suppose $k$ is a line through $M$ parallel to $\ell$. To see why, note that lines $\ell$ and $m$ are parallel by assumption. This task is very closely related to the euclidean parallel postulate.
PARALLEL LINES IN MATH ILLUSTRATIONS SOFTWARE
Students working on this problem will engage in MP5 ''Use Appropriate Tools Strategically'' whether they use geometry software or physical manipulatives. This task provides a good opportunity to explore with geometry software (if available) or with physical manipulatives. Teachers should expect informal arguments as students are only beginning to develop a formal understanding of parallel lines and rigid motions. Students will explore the impact of rotation by 180 degrees on lines in a carefully chosen setting. This goal of this task is to experiment with rigid motions to help visualize why alternate interior angles (made by a transverse connecting two parallel lines) are congruent: this result can then be used to establish that the sum of the angles in a triangle is 180 degrees (see ).
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