Question

1. given ( overline{ab} cong overline{xy} ), and ( o ) is the midpoint of ( overline{ab} ) and of ( overline{xy} ). prove ( overline{xo} cong overline{ao} ).

Explanation:

Step1: Use the mid - point definition

Since \(O\) is the mid - point of \(\overline{AB}\), by the mid - point definition, \(AO=\frac{1}{2}AB\). Since \(O\) is the mid - point of \(\overline{XY}\), \(XO = \frac{1}{2}XY\).

Step2: Use the given congruence

Given \(\overline{AB}\cong\overline{XY}\), then \(AB = XY\) (by the definition of congruent segments: if \(\overline{AB}\cong\overline{XY}\), their lengths are equal).

Step3: Substitute

Substitute \(AB = XY\) into the equations from Step 1. Let \(AB = XY = k\). Then \(AO=\frac{1}{2}k\) and \(XO=\frac{1}{2}k\).

Answer:

Since \(AO=\frac{1}{2}AB\), \(XO=\frac{1}{2}XY\) and \(AB = XY\) (because \(\overline{AB}\cong\overline{XY}\)), we can conclude that \(XO\cong AO\) (by the substitution property of equality for segment lengths, which implies the segments are congruent).