Question

Need help on both of them, help ASAP!!!!!

Answer 1

Parallel lines and congruent angles make corresponding angles in both triangles equal. One side (half of median) and two angles equal, so triangles congruent (AAS). QED (29 words).

Sure, I can help you with this geometry problem. The givens are that $
`\angle ABC \cong \angle CED$ and $\overline{AB} ||`
`\overline{CE}`$. You need to prove that
`\triangle ABC \cong \triangle CED`.

I can see that you have already started working on the proof. You have written down the first three statements and reasons:

1.
`\angle ABC \cong \angle CED`(Given)

2.
`$\overline{AB} || \overline{CE}$`(Given)

3.
`$\angle BAC \cong \angle DCE$` (Vertical angles)

The next step is to use these statements to prove that one of the side pairs of the triangles is congruent. Once you have proven that one pair of sides is congruent, you can use the Angle-Angle-Side (AAS) congruence criterion to prove that the triangles are congruent.

Here are two possible approaches you could take:

Approach 1:Use the fact that
`$\overline{AB}$` is a median of
`$\triangle ADC$` to prove that
`$AC = EC$.`

4.
`$C$` is the midpoint of
`$\overline{AD}$` (Given)

5.
`$AB$` is a median of
`$\triangle ADC$` (Definition of a median)

6.
`$AC = EC$` (Property of a median)

Approach 2: Use the fact that
`$\overline{CE}$` is a transversal to parallel lines
`$\overline{AB}$`and
`$\overline{DC}$` to prove that
`$\angle ACB \cong`
`\angle DCE$.`

4.
`$\overline{CE}$` is a transversal to
`$\overline{AB}$` and
`$\overline{DC}$` (Given)

5.
`$\angle ACB \cong \angle DCE$` (Alternate interior angles)

Once you have proven that one pair of sides is congruent and one pair of angles is congruent, you can use the AAS congruence criterion to conclude that
`$\triangle ABC \cong \triangle CED$.`