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 $\Delta OAB\cong \Delta OCD$, by ASA and AAA method.

Step-by-step solution

- 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.

- Understand the figure

First name the triangles in the given figure,

Here, $\overline{AO}\cong \overline{CO}$ and $\overline{AB}\left|\right|\overline{DC}$.

- Prove the congruency, if possible

Now,

In $\Delta OAB\text{and}\Delta OCD$

$\overline{AO}\cong \overline{CO}$(given)

$\angle AOB\cong \angle COD$(Vertical angles since \[\overline{AB}||\overline{DC}\])

Also,

$\angle OAB\cong \angle OCD\text{and}\angle OBA\cong \angle ODC$(Alternate angles)

As per the definitions of method ASA and AAA, $\Delta OAB\cong \Delta OCD$.

Therefore,$\Delta OAB\cong \Delta OCD$ by ASA and AAA method.