Question

AC and BD bisect eachother prove AB || CD and BC ll AD

Answer 1

1) $\overline{AC}$ and $\overline{BD}$ bisect each other (given)

2) $\overline{AE} \cong \overline{EC}$ (a bisector splits a segment into two congruent parts)

3) $\overline{BE} \cong \overline{ED}$ (a bisector splits a segment into two congruent parts)

4) $\angle BEA \cong \angle CED$ (vertical angles are congruent)

5) $\triangle BEA \cong \triangle DEC$ (SAS)

6) $\angle EBA \cong \angle EDC$ (CPCTC)

7) $\overline{AB} \parallel \overline{CD}$ (converse of alternate interior angles theorem)

8) $\angle DEA \cong \angle BEC$ (vertical angles are congruent)

9) $\triangle AED \cong \triangle BEC$ (SAS)

10) $\angle BCE \cong \angle EAD$ (CPCTC)

11) $\overline{BC} \parallel \overline{AD}$ (converse of alternate interior angles theorem)