Question

what conclusion(s) follow from the given information below?
given: $overline{cd}$ bisects $overline{ab}$ at point $e$.
i. $e$ is the midpoint of $overline{ab}$
ii. $e$ is the midpoint of $overline{cd}$
iii. $overline{ce} \cong \overline{ed}$
iv. $overline{ae} \cong \overline{eb}$
v. $overline{cd} \perp \overline{ab}$
a) i and iv only
b) ii and iii only
c) i, ii, iii, and iv only
d) i, iv and v only
e) v only
f) none of the above.

Explanation:

Step 1: Analyze Conclusion I

A segment bisecting another segment at a point means that point is the midpoint of the bisected segment. Since \( \overline{CD} \) bisects \( \overline{AB} \) at \( E \), \( E \) is the midpoint of \( \overline{AB} \). So Conclusion I is valid.

Step 2: Analyze Conclusion II

There is no information given that \( E \) is related to \( \overline{CD} \) as a midpoint. The given is only about \( \overline{CD} \) bisecting \( \overline{AB} \), so we can't conclude \( E \) is the midpoint of \( \overline{CD} \). Conclusion II is invalid.

Step 3: Analyze Conclusion III

There's no information to support \( \overline{CE} \cong \overline{ED} \) because we don't know if \( E \) is the midpoint of \( \overline{CD} \) (from Step 2). So Conclusion III is invalid.

Step 4: Analyze Conclusion IV

If \( E \) is the midpoint of \( \overline{AB} \) (from Step 1), then by definition of midpoint, \( \overline{AE} \cong \overline{EB} \) (they have equal length). So Conclusion IV is valid.

Step 5: Analyze Conclusion V

There's no information given about the angle between \( \overline{CD} \) and \( \overline{AB} \), so we can't conclude \( \overline{CD} \perp \overline{AB} \). Conclusion V is invalid.

From the above analysis, only Conclusions I and IV are valid, which corresponds to option a) I and IV only.

Answer:

a) I and IV only