Side, Side, Angle Congruence Rule - Examples, Exercises and Solutions

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.

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.

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

Although the lengths of the sides are equal in both triangles, we observe that in the right triangle the angle is adjacent to the side whose length is 7, while in the triangle on the left side the angle is adjacent to the side whose length is 5.

Since it's not the same angle, the angles between the triangles do not match and therefore the triangles are not congruent.

Let's consider that:

DF = AC = 8

DE = AB = 5

8 is greater than 5, therefore the angle DEF is opposite the larger side and is equal to 65 degrees.

That is, the figure we are missing is the angle of the second triangle.

We will examine which angle is opposite the large side AC.

ABC is the angle opposite the larger side AC so it must be equal to 65 degrees.

Let's observe the angle in each of the triangles and note that each time it is opposite to the length of a different side.

Therefore, none of the triangles are congruent since it is impossible to know from the data.

In order for triangles to be congruent, one must demonstrate that the S.A.S theorem is satisfied

We have a common side whose length in both triangles is equal to 3.

Now let's examine the lengths of the other sides:

$2X+4=X+2$

We proceed with the sections accordingly:$2-4=2X-X$

$-2=X$

We place this value in the right triangle we should find the length of the side:$-2+2=0$

However since it is not possible for the length of a side to be equal to 0, the triangles are not congruent.

It is not possible to add data for the triangles to be congruent since the corresponding angles are not equal to each other and therefore the triangles could not be congruent to each other.

To answer the question, we need to know all four congruence theorems -

SAS, SAA, ASA, SSA

Now let's see what data we can prove from the question -

AB is parallel to DC

And from this it follows that angle BAC equals angle ACD, because these are equal alternate angles,

Also, we have a shared side AC, so we have a shared angle and side.

We could have proven congruence using AB=DC, and then we would have SAA,

but this is not given to us in the options,

so let's look at the fourth congruence theorem - SSA,

for it to work, we need to show another side, BC=AD,

and this is indeed one of the options!

But it's important to remember that the fourth congruence theorem has a condition,

the theorem is valid only if the angle is opposite to the larger of the two sides,

therefore we need to know that ACAC,

and indeed, one of these options exists!

Thus, we can see that there are two things we need,

and therefore answer D is correct!