Question

reasoning

1. in the diagram shown, \\( \angle c a b \cong \angle d a b \\). if \\( \angle c a b \\) was reflected over \\( \overrightarrow{a b} \\):

(a) why would \\( \overrightarrow{a c} \\) get mapped on top of \\( \overrightarrow{a d} \\)?
(b) would point \\( c \\) have to get mapped on top of point \\( d \\)? explain.

Explanation:

(a) When a figure is reflected over a line, corresponding angles are congruent. Since \(\angle CAB\cong\angle DAB\), the reflection over \(\overrightarrow{AB}\) maps \(\overrightarrow{AC}\) to \(\overrightarrow{AD}\) because the angles formed with \(\overrightarrow{AB}\) are equal.
(b) A reflection is a rigid transformation. If \(\overrightarrow{AC}\) is mapped to \(\overrightarrow{AD}\) (from part (a)), and \(A\) is a common point (unchanged in a reflection over a line through \(A\)), then the entire ray \(\overrightarrow{AC}\) is mapped to \(\overrightarrow{AD}\). Points on the ray must correspond, so point \(C\) (on \(\overrightarrow{AC}\)) is mapped to point \(D\) (on \(\overrightarrow{AD}\)).

Answer:

(a) Because \(\angle CAB\cong\angle DAB\) (corresponding angles in a reflection).
(b) Yes, because a reflection is a rigid transformation and \(A\) is fixed, and \(\overrightarrow{AC}\) maps to \(\overrightarrow{AD}\).