Proving triangles using ASA and SSS

Question

Given ∠E≅∠C, ∠EDA≅∠CDB, and D is the midpoint of EC¯

Prove ∇DAE≅DBC

                                                                           

Answer 1

First of all draw out the line segment EC and put D in the middle of EC indicating being the midpoint of EC. Than draw out the two triangles DAE and DBC. Notice the two triangles share a common vertex being point D.

 

Now your given ∠E≅∠C and ∠EDA∠≅∠CDB in which are two sets of corresponding angles of each triangle being congruent. Since D is a midpoint of line segment EC, then ED≅CD. Now this would satisfy the ASA proof because you got two angles and a side of each triangle being congruent.

 

To Satisfy the SSS proof, you have to prove that the other sides are congruent. Since there are two angles of each of the triangles are congruent, then the corresponding third angles have to be congruent in which:

 

∠B≅∠A

 

Notice each congruent angle are also alternating interior angles of line segments EA and BC. This would indicate

 

EA is parallel to BC. With ∠E≅∠C, this would indicate EA≅BC. With ∠B≅∠A and∠EDA∠≅∠CDB , this would indicate BD≅DA in which proves D is the midpoint of line segment AB. Therefore, with three sides being congruent, this would satisfy the SSS proof.

 

Note: I'll have to review the theorems and postulates to put together the proof in a nice format.

 

I did some digging. Here's the proof:

 

Statements          | Reasons____

∠E≅∠C               |  Given

∠EDA∠≅∠CDB    |  Given

ED≅CD               |  Definition of Bisect

∇DEA≅∇DBC      | ASA

∠B≅∠A               | Vertical Angles are congruent

BD≅AD               | BD and AD bisects AB

BC≅EA                | Corresponding sides are congruent

∇DEA≅∇DBC      | SSS