Question

statements reasons

1. 1. given
2. ( ac = ce )

( bc = cd ) 2.

1. ( ae = bd ) 3.
2. 4. segment addition

property

1. ( ac + ce = bc + cd ) 5.
2. ( ac + ac = cd + cd ) 6.
3. 7. simplify
4. 8. division property of

equality

1. ( overline { a c } cong overline { c d } ) 9.

Explanation:

Step1: Identify the first given statement

Since \(AC = CE\) and \(BC = CD\) are given later, the first given statement is likely \(AE=BD\) (assuming the problem starts with \(AE = BD\) as given).

Step2: Reason for \(AC = CE\) and \(BC = CD\)

If we assume some mid - point or congruent segment definitions (not fully clear from the given partial table), but if we consider the structure of the proof, if \(E\) is the mid - point of \(AC\) (so \(AC = CE\)) and \(D\) is the mid - point of \(BC\) (so \(BC = CD\)), but more likely, if we consider the property of equality of segments. If we assume that \(AC = CE\) and \(BC = CD\) are given (but the first “Given” is for \(AE = BD\)), there is a mis - alignment. Let's re - approach.

Let's assume the first statement is \(AE=BD\) (Reason: Given).

For \(AC = CE\) and \(BC = CD\), if we assume that \(E\) is the mid - point of \(AC\) and \(D\) is the mid - point of \(BC\), but more likely, if we consider the property of equality of segments. Let's use substitution.

If \(AE=BD\) (Given), and by the Segment Addition Property \(AE=AC + CE\) and \(BD=BC + CD\).

Step3: Reason for \(AE = BD\) (if it's the first statement)

If \(AE = BD\) is the first statement, the reason is “Given”. But if we assume the first “Given” is for \(AC = CE\) and \(BC = CD\) (which is not standard), there is an error. Let's assume the first statement is \(AE=BD\) (Reason: Given)

For \(AC + CE=BC + CD\), by substitution (since \(AE = AC + CE\), \(BD=BC + CD\) and \(AE = BD\))

For \(AC + AC=CD + CD\), by substitution (since \(AC = CE\) and \(BC = CD\))

For \(2AC=2CD\) (Step 7: Simplify \(AC + AC=CD + CD\))

For \(AC = CD\) (Step 8: By Division Property of Equality, divide both sides of \(2AC = 2CD\) by 2)

For \(\overline{AC}\cong\overline{CD}\) (Step 9: If two segments have equal lengths (\(AC = CD\)), then they are congruent (Definition of Congruent Segments))

Answer:

1. \(AE = BD\)
2. Given (assuming a mis - order in the table, if \(AC = CE\) and \(BC = CD\) are given) or Mid - point (if \(E\) is mid - point of \(AC\) and \(D\) is mid - point of \(BC\))
3. Given (if \(AE = BD\) is in a wrong row) or Segment Addition (if \(AE=AC + CE\) and \(BD=BC + CD\))
4. \(AE=AC + CE\), \(BD = BC + CD\)
5. Substitution (since \(AE = BD\))
6. Substitution (since \(AC = CE\) and \(BC = CD\))
7. \(2AC=2CD\)
8. \(AC = CD\)
9. Definition of Congruent Segments