Question

consider the following geometric proof. given: o is the midpoint of \\(\overline{ac}\\) and \\(\overline{bd}\\). prove: \\(\triangle aod \cong \triangle boc\\) \
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statements	reasons	\
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1. o is the midpoint of \\(\overline{ac}\\) and \\(\overline{bd}\\).	1. given	\
2. \\(\overline{ao} \cong \overline{co}; \overline{bo} \cong \overline{do}\\)	2. the segments look congruent	\
3. \\(\angle aod \cong \angle boc\\)	3. given	\
4. \\(\triangle aod \cong \triangle boc\\)	4. sas	\

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select the correct answer from each drop - down menu.\
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reason number 1 is valid because it \\(\boldsymbol{}\\), reason number 2 is invalid because \\(\boldsymbol{}\\), reason number 3 is \\(\boldsymbol{}\\), reason number 4 can be used only if \\(\boldsymbol{}\\).

Explanation:

1. Reason 1 is valid because it directly restates the given premise provided in the proof setup.
2. Reason 2 is invalid because visual appearance ("segments look congruent") is not a valid geometric proof reason; midpoints define congruent segments by definition, not visual judgment.
3. Reason 3 is invalid because ∠AOD ≅ ∠BOC is vertical angles, not a given statement.
4. Reason 4 (SAS) can be used only if we have two pairs of congruent sides and the included congruent angle, which requires valid reasons for the side and angle congruence first.

Answer:

Reason number 1 is valid because it matches the given premise, Reason number 2 is invalid because visual appearance is not a proof, Reason number 3 is invalid (it is vertical angles, not given), Reason number 4 can be used only if we have valid side/angle congruence justifications.