can you match the teachers comments to the definitions?
definition of perpendicular lines teachers comments
danielas definition: $ \overleftrightarrow { a b } $ is
perpendicular to $ \overleftrightarrow { c d } $ if the
measure of $ \angle a o c $ equals the
measure of $ \angle a o d $. please rework your definition to use
more precise mathematical language.
oris definition: $ \overleftrightarrow { a b } $ is
perpendicular to $ \overleftrightarrow { c d } $ if $ \overleftrightarrow { a b } $
and $ \overleftrightarrow { c d } $ intersect perfectly
sidways. well done! i hadnt previously
considered defining perpendicular
lines in terms of a rigid
transformation like rotation, but by
golly it works!
kaoris definition: $ \overleftrightarrow { a b } $ is
perpendicular to $ \overleftrightarrow { c d } $ if one
line can be mapped to the
other by a rotation about point
$ o $ of $ 90 ^ { \circ } $. well done! because the measure of
$ \angle c o d $ is $ 180 ^ { \circ } $ (by the definition of
a straight angle), the measures of
angles $ \angle a o c $ and $ \angle a o d $ must
both be $ 90 ^ { \circ } $.

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# Brief Explanations:
- Daniela's definition: The statement about angles $ \angle AOC $ and $ \angle AOD $ having equal measures is a valid geometric concept related to perpendicular lines (since if two adjacent angles formed by two intersecting lines are equal and their sum is $ 180^{\circ}$ (as $ \angle COD = 180^{\circ}$ by the definition of a straight - angle), then each angle is $ 90^{\circ}$). So, the comment “Well done! Because the measure of $ \angle COD$ is $ 180^{\circ}$ (by the definition of a straight angle), the measures of angles $ \angle AOC$ and $ \angle AOD$ must both be $ 90^{\circ}$” matches Daniela's definition.
- Ori's definition: The phrase “intersect perfectly sideways” is not precise mathematical language. So, the comment “Please rework your definition to use more precise mathematical language” matches Ori's definition.
- Kaori's definition: A rotation of $ 90^{\circ}$ about a point $ O$ is a rigid transformation (a type of transformation in geometry). So, the comment “Well done! I hadn't previously considered defining perpendicular lines in terms of a rigid transformation like rotation, but by golly it works!” matches Kaori's definition.

# Answer:
- Daniela's definition matches “Well done! Because the measure of $ \angle COD$ is $ 180^{\circ}$ (by the definition of a straight angle), the measures of angles $ \angle AOC$ and $ \angle AOD$ must both be $ 90^{\circ}$”.
- Ori's definition matches “Please rework your definition to use more precise mathematical language”.
- Kaori's definition matches “Well done! I hadn't previously considered defining perpendicular lines in terms of a rigid transformation like rotation, but by golly it works!”.

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