Question

complete the proof that the diagonals of parallelogram abcd bisect each other.
this proof uses the following theorem: opposite sides of a parallelogram are congruent.
statement reason
1 $overline{ab} parallel overline{cd}$ and $overline{ad} parallel overline{bc}$ given
2 $angle 1 cong angle 3$ when a transversal crosses parallel lines, alternate interior angles are congruent. (i)
3 $angle 2 cong angle 4$ when a transversal crosses parallel lines, alternate interior angles are congruent. (i)
4 $overline{ab} cong overline{cd}$ opposite sides of a parallelogram are congruent. (i)
5 pick statement pick congruence criterion congruence
6 pick statement corresponding parts of congruent triangles are congruent. (i)
7 point e bisects both $overline{ac}$ and $overline{bd}$ definition of bisector (iii)

Explanation:

Step1: Identify Triangles to Prove Congruent

We have parallelogram \(ABCD\) with diagonals intersecting at \(E\). From steps 2, 3, and 4, we know \(\angle1\cong\angle3\), \(\angle2\cong\angle4\), and \(\overline{AB}\cong\overline{CD}\). So, we can prove \(\triangle ABE\cong\triangle CDE\) (or \(\triangle ADE\cong\triangle CBE\)) using the ASA (Angle - Side - Angle) congruence criterion. The statement for step 5 should be \(\triangle ABE\cong\triangle CDE\) (or the other pair) and the reason is ASA (since we have two angles and the included side congruent: \(\angle1\cong\angle3\), \(\overline{AB}\cong\overline{CD}\), \(\angle2\cong\angle4\)).

Step2: Corresponding Parts of Congruent Triangles

After proving the triangles congruent, the corresponding parts (like \(\overline{AE}\cong\overline{CE}\) and \(\overline{BE}\cong\overline{DE}\)) are congruent by CPCTC (Corresponding Parts of Congruent Triangles are Congruent). So the statement for step 6 would be \(\overline{AE}\cong\overline{CE}\) and \(\overline{BE}\cong\overline{DE}\).

Answer:

Step 5:

• Statement: \(\triangle ABE\cong\triangle CDE\) (or \(\triangle ADE\cong\triangle CBE\))
• Reason: ASA (Angle - Side - Angle) Congruence Criterion (because \(\angle1\cong\angle3\), \(\overline{AB}\cong\overline{CD}\), \(\angle2\cong\angle4\))

Step 6:

• Statement: \(\overline{AE}\cong\overline{CE}\) and \(\overline{BE}\cong\overline{DE}\)
• Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)