Question

proof in the figure, point b is the midpoint of (overline{ac}) and point c is the midpoint of (overline{bd}). complete the paragraph proof to prove that (ab = cd)
given: (b) is the midpoint of (overline{ac}). (c) is the midpoint of (overline{bd}).
prove: (ab = cd)
image: line segment with points (a), (b), (c), (d) in order
proof: because (b) is the midpoint of (overline{ac}) and (\boldsymbol{\text{select choice}}) is the midpoint of (overline{bd}), we know by the (\boldsymbol{\text{select choice}}) that (overline{ab} cong overline{bc}) and (overline{bc} cong overline{cd}). because congruent segments have (\boldsymbol{\text{select choice}}) measures, (ab = bc) and (\boldsymbol{\text{select choice}}). thus, by the (\boldsymbol{\text{select choice}}), (ab = cd).

Explanation:

Step1: Identify the midpoint of \( \overline{BD} \)

Given that \( C \) is the midpoint of \( \overline{BD} \), so the first "Select Choice" should be \( C \).

Step2: Recall the Midpoint Definition

The midpoint of a segment divides it into two congruent segments. This is the definition of a midpoint, so the second "Select Choice" is "Definition of Midpoint".

Step3: Congruent Segments Property

Congruent segments have equal measures, so the third "Select Choice" is "equal".

Step4: Substitute \( BC \) with \( CD \)

Since \( \overline{BC} \cong \overline{CD} \), we know \( BC = CD \) (from the equal measures of congruent segments). So the fourth "Select Choice" is \( BC = CD \).

Step5: Transitive Property of Equality

If \( AB = BC \) and \( BC = CD \), then by the Transitive Property of Equality, \( AB = CD \). So the fifth "Select Choice" is "Transitive Property of Equality".

Answer:

1. First "Select Choice": \( C \)
2. Second "Select Choice": Definition of Midpoint
3. Third "Select Choice": equal
4. Fourth "Select Choice": \( BC = CD \)
5. Fifth "Select Choice": Transitive Property of Equality