Question

self - assessment

1. complete the two - column proof. then write a flowchart proof.

given: ab = de, bc = cd
prove: ac = ce
statements

1. ab = de, bc = cd
2. ab + bc = bc + de

3.

1. ab + bc = ac, cd + de = ce

5.

1. ac = ce

reasons

1. given
2. addition property of equality
3. substitution property of equality

4.

1. substitution property of equality

6.

1. write a two - column proof of the vertical angle congruence theorem without using the congruence theorem. compare your proof with the proof in example 3.

statements reasons
chapter 2 reasoning and proofs

Explanation:

Step1: Substitute equal - length segments

Since \(AB = DE\) and \(BC=CD\), we substitute \(DE\) for \(AB\) and \(CD\) for \(BC\) in the equation \(AB + BC=BC + DE\). So, the statement for step 3 is \(AB + BC=CD + DE\).

Step2: Recall segment - addition postulate

The reason for statement 4 (\(AB + BC = AC\) and \(CD + DE=CE\)) is the Segment - Addition Postulate which states that if we have two adjacent segments, the length of the combined segment is the sum of the lengths of the two smaller segments.

Step3: Substitute segment sums

Substitute \(AC\) for \(AB + BC\) and \(CE\) for \(CD + DE\) in the equation from step 3. So, the statement for step 5 is \(AC = CE\).

Step4: Use definition of congruent segments

If two segments have the same length, they are congruent. So the reason for step 6 (\(\overline{AC}=\overline{CE}\)) is the Definition of Congruent Segments.

Answer:

1. \(AB + BC=CD + DE\)
2. Segment - Addition Postulate
3. \(AC = CE\)
4. Definition of Congruent Segments