Question

write a proof
given c is the midpoint of ae and bd.
prove △abc ≅ △edc
statements
c is the midpoint of ae and bd
reasons
given
vertical angles congruence theorem
definition of midpoint
△abc ≅ △edc
ac ≅ cd, bc ≅ ce
ac ≅ ec, bc ≅ dc
ac ≅ bc, dc ≅ ec
∠acb ≅ ∠ecd
∠abc ≅ ∠cde
∠bac ≅ ∠ced
△abc ≅ △edc
△abc ≅ △ced
△bac ≅ △dce
vertical angles congruence theorem
alternate interior angles theorem
sas congruence theorem

Explanation:

Step1: Use the mid - point definition

Since \(C\) is the mid - point of \(AE\) and \(BD\), we have \(AC = CE\) and \(BC=CD\) (by the definition of a mid - point).

Step2: Use the vertical angles congruence

\(\angle ACB\) and \(\angle ECD\) are vertical angles. By the Vertical Angles Congruence Theorem, \(\angle ACB\cong\angle ECD\).

Step3: Use the SAS (Side - Angle - Side) congruence criterion

In \(\triangle ABC\) and \(\triangle EDC\), we have:

• \(AC = CE\) (from Step 1)
• \(\angle ACB=\angle ECD\) (from Step 2)
• \(BC = CD\) (from Step 1)

By the SAS congruence criterion, \(\triangle ABC\cong\triangle EDC\).

Answer:

\(\triangle ABC\cong\triangle EDC\) (by the SAS congruence criterion)