Proof:
• Construct an auxiliary line through point

*C* bisecting ∠

*C*. An angle has a unique angle bisector. Label the intersection with the base as

*D*.

•

*m∠ACD = m∠BCD* because an angle bisector forms two congruent angles which have equal measure.

• Under a reflection in

, the reflection of

*C* will be

*C*, since

*C* lies on the line of
reflection.

• Since

*m∠ACD = m∠BCD* and reflections preserve angle measure, the image of ∠

*ACD* will be the same measure as ∠

*BCD.*
• Since these angles are equal in measure, the reflection of ray

(side of the ∠) will coincide with its image

(side of the image angle).

• The reflection of

will have the same length as that of

since reflections preserve length.

• The reflection of

will have the same length as that of

by substitution.

• The reflection of

*A* is

*B* since reflections preserve length and the segments share point

*C*.

• The reflections of

and the reflection of

since reflections map rays to rays.

• The reflection of ∠

*CAB* will have the same measure as ∠

*CBA* since reflections preserve angle measure. We have established that the rays forming these angles coincide under a reflection.

•

since congruent angles are angles of equal measure.

QED