Elena is thinking through a proof using a reflection to show that the base angles of an isosceles triangle are congruent. Complete the missing information for her proof. 

Call the midpoint of segment  . Construct the perpendicular bisector of segment . The perpendicular bisector of must go through since it's the midpoint.  is also on the perpendicular bisector of because the distance from to  is the same as the distance from to . We want to show triangle is congruent to triangle . Reflect triangle across line . Since is on the line of reflection, it definitely lines up with itself. is congruent to  since is the perpendicular bisector of . will coincide with since it is on the other side of a perpendicular line and the same distance from it (and that’s the definition of reflection!). will coincide with since it is on the other side of a perpendicular line and the same distance from it (and that’s the definition of reflection!). Since the rigid transformation will take triangle  onto triangle , that means angle will be taken onto angle (they are corresponding parts under the same reflection). Therefore, they are congruent.

Segment  is an angle bisector of angle . Noah wrote a proof to show that triangle  is congruent to triangle . Noah's proof is not correct. Why is Noah's proof incorrect?

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• Side  is congruent to side  because they're the same segment.
• Angle  is congruent to angle  because segment  is an angle bisector of angle .
• Angle  is congruent to angle  because segment  is an angle bisector of angle .
• By the Angle-Side-Angle Triangle Congruence Theorem, triangle  is congruent to triangle .