Conditions of Congruency: Identifying and defining elements

Let's consider that:

AC=EF=4

DF=AB=5

Since 5 is greater than 4 and the angle equal to 34 is opposite the larger side in both triangles, the angle ACB must be equal to the angle DEF

Therefore, the triangles are congruent according to the SAS theorem, as a result of this all angles and sides are congruent, and all answers are correct.

According to the existing data:

$EF=BA=10$(Side)

$ED=AC=13$(Side)

The angles equal to 53 degrees are both opposite the greater side (which is equal to 13) in both triangles.

(Angle)

Since the sides and angles are equal among congruent triangles, it can be determined that angle DEF is equal to angle BAC

This question actually has two steps:

In the first step, you must define if the triangles are congruent or not,

and then identify the correct answer among the options.

Let's look at the triangles: we have two equal sides and one angle,

But this is not a common angle, therefore, it cannot be proven according to the S.A.S theorem

Remember the fourth congruence theorem - S.A.A
If the two triangles are equal to each other in terms of the lengths of the two sides and the angle opposite to the side that is the largest, then the triangles are congruent.

But the angle we have is not opposite to the larger side, but to the smaller side,

Therefore, it is not possible to prove that the triangles are congruent and no theorem can be established.

In answer A, we are given two triangles with different angles, therefore the sides are also different and they are not congruent according to S.S.S.

In answer B, we are given two right triangles, but their angles are different and so are the sides. Therefore, they are not congruent according to S.S.S.

In answer D, we do not have enough data, therefore it is not possible to determine that they are congruent according to S.S.S.

In answer C, we see that all the sides are equal to each other in both triangles and therefore they are congruent according to S.S.S.

Let's consider the corresponding congruent triangles letters:

$CBO=ABO$

That is, from this we can determine:

$CB=AB$

$BO=BO$

$CO=AO$

We observe the order of the letters in the congruent triangles and write the matches (from left to right).

$ABC=CDA$

That is:

Angle A is equal to angle C.

Angle B is equal to angle D.

Angle C is equal to angle A.

From this, it is deduced that angle BAC (where the letter A is in the middle) is equal to angle C — that is, to angle DCA (where the letter C is in the middle).

There are similar triangles that are not necessarily congruent, so this statement is not correct.