Question

proving segment relationships · practice
example 1

1. proof write the correct statements and reasons to complete the two - column proof.

given: c is the midpoint of \\( \overline { a e } \\).
c is the midpoint of \\( \overline { b d } \\).
\\( \overline { a e } \cong \overline { b d } \\)
prove: \\( \overline { a c } \cong \overline { c d } \\)
proof:

1. proof write the correct statements and reasons to complete the two - column proof.

given: \\( \overline { s u } \cong \overline { l r } \\)
\\( \overline { t u } \cong \overline { l n } \\)
prove: \\( \overline { s t } \cong \overline { n r } \\)
proof:

Explanation:

Step1: Fill in the first proof

Statements

1. \(C\) is the midpoint of \(\overline{AE}\), \(C\) is the midpoint of \(\overline{BD}\), \(\overline{AE}\cong\overline{BD}\)
2. \(AE = AC + CE\), \(BD=BC + CD\)
3. \(2AC = 2CD\)
4. \(AC = CD\)

Reasons

1. Definition of mid - point
2. Definition of congruent segments
3. Substitution Property
4. Substitution Property (using \(AC = CE\) and \(BC = CD\))
5. Definition of congruent segments

Step2: Fill in the second proof

Statements

1. \(SU = LR\), \(TU = LN\)
2. \(ST=NR\)

Reasons

1. Given
2. Segment Addition Property
3. Substitution Property (using \(SU = LR\) and \(SU = ST + TU\), \(LR = LN + NR\))
4. Substitution Property (using \(TU = LN\))
5. Subtraction Property of Equality
6. Definition of congruent segments

Answer:

First proof:

• Statements:

1. \(C\) is the midpoint of \(\overline{AE}\), \(C\) is the midpoint of \(\overline{BD}\), \(\overline{AE}\cong\overline{BD}\)
2. \(AE = AC + CE\), \(BD=BC + CD\)
3. \(2AC = 2CD\)
4. \(AC = CD\)

• Reasons:

1. Definition of mid - point
2. Definition of congruent segments
3. Substitution Property
4. Substitution Property
5. Definition of congruent segments

Second proof:

• Statements:

1. \(SU = LR\), \(TU = LN\)
2. \(ST=NR\)

• Reasons:

1. Given
2. Segment Addition Property
3. Substitution Property
4. Substitution Property
5. Subtraction Property of Equality
6. Definition of congruent segments