Question

Can two triangles be proved congruent? If so, by which method, SSS, SAS, ASA, AAS, or HL?

Short answer

Yes, the two triangles can be proved congruent $\Delta EFG\cong \Delta ECD$, by SAS method.

Step-by-step solution

Step 1.  Define methods SSS, SAS, ASA, AAS, and HL

SSS: Side–Side–Side method means if three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

SAS: Side – Angle–side method meansif two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, then triangles are congruent.

ASA: Angle–side–angle method means if two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, then triangles are congruent.

AAS: Angle–angle–side method if two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, then triangles are congruent.

HL: Hypotenuse–leg method means if the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, then two triangles are congruent.

Step 2.  Understand the figure

First name the triangles in the given figure,

Here, $\overline{ED}\cong \overline{EG}$and $\overline{DF}\cong \overline{GC}$.

Step 3.  Prove the congruency, if possible

In $\Delta EFG\text{and}\Delta ECD$,

i. It is given that, $\overline{ED}\cong \overline{EG}$ and $\overline{DF}\cong \overline{GC}$.

ii. From (i), $\overline{ED}+\overline{DF}=\overline{EG}+\overline{GC}$, by Segment addition postulate, $\overline{EF}\cong \overline{EC}$.

iii. By Reflexive Property, $\overline{E}\cong \overline{E}$.

Yes, the above two triangles $\Delta EFG\cong \Delta ECD$ are congruent by SAS postulate.