Conditions of Congruency: Ascertaining whether or not the triangles are congruent

Congruent triangles are triangles that are identical in size, so if we place one on top of the other, they will match exactly.

To prove that a pair of triangles are congruent, we need to prove that they satisfy one of these three conditions:

1. SSS - Three sides of both triangles are equal in length.

2. SAS - Two sides are equal between the two triangles, and the angle between them is equal.

3. ASA - Two angles in both triangles are equal, and the side between them is equal.

If we take an initial look at the drawing, we can already see that there is one equal side between the two triangles (marked in blue),

We don't have information about the other sides, so we can rule out the first two conditions,

And now we'll focus on the last condition - angle, side, angle.

We can see that angle D equals angle A, both equal to 50 degrees,

And now we're focusing on angles E.

At first glance, we might think there's no way to know if these angles are equal, but if we look at how the triangles are positioned,
We can see that these angles are actually corresponding angles, and corresponding angles are equal.

Therefore - if the angle, side, and second angle are equal, we can prove that the triangles are equal using the ASA condition

Since we know that the triangle is isosceles, we can establish that AC=AB and that

AD=AD since it is a common side to the triangles ADC and ADB

Furthermore given that the line AD intersects side BC, we can also establish that BD=DC

Therefore, the triangles are congruent according to the SSS (side, side, side) theorem

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.

To answer the question, we need to know the fourth congruence theorem: S.A.S.

The theorem states that triangles are congruent when they have an equal pair of sides and an equal angle.

However, there is one condition: the angle must be opposite the longer side of the triangle.

We start with the sides:

DF = CB = 16
GD = AC = 9

Now, we look at the angles:

A = G = 120

We know that an angle of 120 is an obtuse angle and this type of angle is always opposite the larger side of the triangle.

Therefore, we can argue that the triangles are congruent according to the S.A.S. theorem.

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.