Question

statements

1. \\( \overline { a e } \\) bisects \\( \overline { b d } \\)
2. \\( \overline { b c } \cong \overline { d c } \\)
3. \\( \angle a \cong \angle e \\)
4. \\( \angle a c b \cong \angle e c d \\)
5. \\( \delta a b c \cong \delta e d c \\)

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Explanation:

Step1: Definition of Bisector

If a line bisects another line, it divides it into two equal parts. So, if \(\overline{AE}\) bisects \(\overline{BD}\), then \(BC = DC\) (by the definition of a bisector of a line segment). So the reason for statement 1 is "Def. of Bisector".

Step2: Vertical Angles

\(\angle ACB\) and \(\angle ECD\) are vertical angles. Vertical angles are formed when two lines intersect. By the vertical - angles theorem, \(\angle ACB\cong\angle ECD\). So the reason for statement 4 is "Vertical Angles".

Step3: AAS (Angle - Angle - Side)

We know that \(\angle A\cong\angle E\) (given), \(\angle ACB\cong\angle ECD\) (vertical angles), and \(BC\cong DC\) (from the bisector definition). The AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent. So the reason for statement 5 is "AAS".

Answer:

1. Def. of Bisector
2. Def. of Bisector (since \(\overline{AE}\) bisects \(\overline{BD}\), \(BC = DC\) which is equivalent to \(\overline{BC}\cong\overline{DC}\))
3. Given
4. Vertical Angles
5. AAS