Question

step statement reason
1
\\( \angle a \cong \angle c \\)
\\( \overline { a e } \cong \overline { c d } \\)
\\( \angle b e d \cong \angle b d e \\)
given
2 \\( \angle a e b \\) and \\( \angle b e d \\) are supplementary if two angles form a linear pair, then they are
supplementary
3 \\( \angle c d b \\) and \\( \angle b d e \\) are supplementary if two angles form a linear pair, then they are
supplementary
4 \\( \angle a e b \cong \angle c d b \\) if two angles are supplements of the same angle (or
congruent angles), then they are congruent
5 \\( \triangle a e b \cong \triangle c d b \\) aas

answer attempt 1 out of 5
the proof is incorrect and step number is the first unjustified step
due to a

Explanation:

Step1: Analyze step 4

In step 4, it is claimed that \(\angle AEB\cong\angle CDB\) because they are supplements of congruent angles (\(\angle BED\cong\angle BDE\)). But for the "supplements of congruent angles are congruent" theorem, we need to have two pairs of angles where \(\angle AEB+\angle BED = 180^{\circ}\) and \(\angle CDB+\angle BDE=180^{\circ}\). Since \(\angle BED\cong\angle BDE\), by the subtraction property of equality (or the congruent supplements theorem), \(\angle AEB\cong\angle CDB\) is justified.

Step2: Analyze step 5

For the AAS (Angle - Angle - Side) congruence criterion, we need two angles and a non - included side of one triangle to be congruent to the corresponding two angles and non - included side of another triangle. We know \(\angle A\cong\angle C\), \(\angle AEB\cong\angle CDB\), and \(AE\cong CD\). But in the AAS criterion, the side should be non - included. However, we have not established that the triangles \(\triangle AEB\) and \(\triangle CDB\) have the correct correspondence of angles and side. The first unjustified step is step 5.

Answer:

5