# Congruent Triangles - Examples, Exercises and Solutions

### Congruent triangles are identical triangles.

That means in triangles whose angles and sides are equal, their area and perimeter will also be equal.

But keep in mind that this case is different from when triangles are similar, that is, when the angles are equal but the side lengths are different in the corresponding ratio.

### Congruence Criteria

To prove that 2 triangles are congruent we can use one of the following criteria:

By verifying one of the triangle congruence criteria, we can affirm that the triangles are congruent.

## Examples with solutions for Congruent Triangles

### Exercise #1

Look at the triangles in the diagram.

Which of the following statements is true?

### Step-by-Step Solution

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.

It is not possible to calculate.

### Exercise #2

Look at the triangles in the diagram.

Determine which of the statements is correct.

### Step-by-Step Solution

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.

All of the above.

### Exercise #3

Look at the triangles in the diagram.

Which of the following statements is true?

### Step-by-Step Solution

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

Angles BAC is equal to angle DEF.

### Exercise #4

Given: ΔABC isosceles

and the line AD cuts the side BC.

And if so, according to which congruence theorem?

### Step-by-Step Solution

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

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.

### Exercise #5

Choose the pair of triangles that are congruent according to S.S.S.

### Step-by-Step Solution

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.

### Exercise #6

Are similar triangles necessarily congruent?

### Step-by-Step Solution

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

No

### Exercise #7

What data must be added so that the triangles are congruent?

### Step-by-Step Solution

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.

Angle ABC equals 65.

### Exercise #8

Triangles ABC and CDA are congruent.

Which angle is equal to angle BAC?

### Step-by-Step Solution

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).

C

### Exercise #9

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

### Step-by-Step Solution

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.

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.

### Exercise #10

Which of the triangles are congruent?

### Step-by-Step Solution

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.

### Exercise #11

The triangles ABO and CBO are congruent.

Which side is equal to BC?

### Step-by-Step Solution

Let's consider the corresponding congruent triangles letters:

$CBO=ABO$

That is, from this we can determine:

$CB=AB$

$BO=BO$

$CO=AO$

Side AB

### Exercise #12

Are the triangles in the image congruent?

If so, according to which theorem?

### Step-by-Step Solution

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.

### Exercise #13

What data must be added so that the triangles are congruent?

### Step-by-Step Solution

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.

Data cannot be added for the triangles to be congruent.

### Exercise #14

Are the triangles in the drawing congruent?

### Step-by-Step Solution

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

### Exercise #15

According to which theorem are the triangles ΔADB≅ΔCBD congruent?