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

Given: ΔABC isosceles

and the line AD cuts the side BC.

Are ΔADC and ΔADB congruent?

And if so, according to which congruence theorem?

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

Congruent by L.L.L.

Are the triangles shown in the diagram congruent? If so, according to which congruence theorem?

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.

Congruent according to S.A.S.

Which of the triangles are congruent?

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.

It is not possible to know based on the data.

Are the triangles in the image congruent?

If so, according to which 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.

No.

Are the triangles in the drawing congruent?

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.

No

Below are two right triangles.

According to which theorem are they congruent?

It can be verified using all theorems.