Question

statements reasons

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

( bc = cd ) 2. given

1. ( ae = bd ) 3. given
2. ( ae = ac + ce ) ( bd = bc + cd ) 4. segment addition

property

1. ( ac + ce = bc + cd ) 5. substitution property
2. ( ac + ac = cd + cd ) 6. substitution property

7 ( 2ac = 2cd ) 7. simplify

1. ( ac = cd ) 8. division property of

equality

1. ( overline { ac } cong overline { cd } ) 9 definition of congruent segments

Explanation:

Step1: Analyze the given information

We are given some equalities and need to follow the logical steps of the proof.

Step2: Use substitution property

Since \(AC = CE\) and \(BC = CD\), in step 6, we substitute \(CE\) with \(AC\) and \(BC\) with \(CD\) in the equation \(AC + CE=BC + CD\) (from step 5) to get \(AC + AC=CD + CD\).

Step3: Simplify the equation

For the equation \(AC + AC=CD + CD\), using the rule of combining like - terms (\(a + a=2a\)), we get \(2AC = 2CD\) (step 7).

Step4: Apply the division property of equality

Dividing both sides of the equation \(2AC = 2CD\) by 2 (since if \(a=b\), then \(\frac{a}{c}=\frac{b}{c}\), \(c
eq0\)), we get \(AC = CD\) (step 8).

Step5: Use the definition of congruent segments

If two segments have the same length (\(AC = CD\)), then by the definition of congruent segments (\(\overline{AB}\cong\overline{CD}\) if and only if \(AB = CD\)), we have \(\overline{AC}\cong\overline{CD}\) (step 9).

Answer:

The proof shows that \(\overline{AC}\cong\overline{CD}\) through a series of substitution, simplification, and application of equality properties.