Question

proof complete the two - column proof. the statements are provided on the left. drag the reasons to the corresponding statements.
given: c is the midpoint of (overline{ae}).
c is the midpoint of (overline{bd}).
(overline{ae}congoverline{bd})
prove: (overline{ac}congoverline{cd})
proof:

Explanation:

Step1: Use the mid - point property

Since \(C\) is the mid - point of \(\overline{AE}\), by the definition of a mid - point, \(AC = CE\). Also, since \(C\) is the mid - point of \(\overline{BD}\), \(BC=CD\).

Step2: Use the vertical - angles and congruent - segments property

We know that \(\angle ACB=\angle ECD\) (vertical angles are equal). Given \(\overline{AE}\cong\overline{BD}\), then \(AC + CE=BC + CD\). Since \(AC = CE\) and \(BC = CD\), and \(AC + CE=BD\), \(BC + CD = AE\). Substituting \(CE\) for \(AC\) (because \(AC = CE\)) and \(CD\) for \(BC\) (because \(BC = CD\)) in the equation \(AC + CE=BC + CD\), we get \(2AC=2CD\).

Step3: Simplify the equation

Dividing both sides of the equation \(2AC = 2CD\) by \(2\), we get \(AC=CD\). By the definition of congruent segments (if two segments have the same length, they are congruent), \(\overline{AC}\cong\overline{CD}\).

Answer:

\(\overline{AC}\cong\overline{CD}\) is proved as shown above.