Question

The figure below shows rectangle ABCD with diagonals segment AC and segment DB:

A rectangle ABCD is shown with diagonals AC and BD.

Jimmy wrote the following proof to show that the diagonals of rectangle ABCD are congruent:

Jimmy's proof:
Statement 1: Rectangle ABCD is given
Statement 2: segment AB ≅ segment DC because opposite sides of a rectangle are congruent
Statement 3: Angles ABC and DCB are both right angles by definition of a rectangle
Statement 4: Angles ABC and DCB are congruent because all right angles are congruent)
Statement 5:
Statement 6: Triangles ABC and DCB are congruent by SAS
Statement 7: segment AC ≅ segment DB by CPCTC

Which statement below completes Jimmy's proof?

segment AD ≅ segment AD (reflexive property of congruence)
segment AD ≅ segment AD (transitive property of congruence)
segment BC ≅ segment BC (reflexive property of congruence)
segment BC ≅ segment BC (transitive property of congruence)

Answer 1

The missing statement in Jimmy's proof is: Statement 5: segment AD ≅ segment AD (reflexive property of congruence).

The reflexive property of congruence states that any segment is congruent to itself. This means that segment AD is congruent to segment AD, regardless of any other information about the figure.

Once Jimmy establishes that segments AD and AD are congruent, he can then use the SAS congruence postulate to prove that triangles ABC and DCB are congruent. This is because he has already shown that these triangles have two congruent sides and a congruent angle.

Therefore, the correct answer is segment AD ≅ segment AD (reflexive property of congruence).

The other answer choices are incorrect:

Segment AD ≅ segment AD (transitive property of congruence): The transitive property of congruence states that if segment A is congruent to segment B, and segment B is congruent to segment C, then segment A is congruent to segment C. However, Jimmy has not yet established that segment AD is congruent to any other segment.

Segment BC ≅ segment BC (reflexive property of congruence): The reflexive property of congruence applies to any segment, including segment BC. However, it is not necessary to prove this statement in order to complete Jimmy's proof.

Segment BC ≅ segment BC (transitive property of congruence): The transitive property of congruence can be used to prove that segment BC is congruent to itself. However, this is not necessary to complete Jimmy's proof.