Congruence of Right Triangles (using the Pythagorean Theorem) - Examples, Exercises and Solutions

In right triangles, we have a condition that already exists in the first place. It refers to the right angle that iss given and that turns a triangle into a right triangle.

In the second stage, we will move on to the sides. In every right triangle we have two perpendiculars (two sides between which the right angle is comprised) and the other (the larger side of the triangle that faces the right angle).

When there are two right triangles in front of us, in which one size is perpendicular and the size of the rest is equal to each other, then we can conclude that these are congruent triangles.

Practice Congruence of Right Triangles (using the Pythagorean Theorem)

Exercise #1

Look at the triangles in the diagram.

Which of the following statements is true?

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Step-by-Step Solution

According to the existing data:

EF=BA=10 EF=BA=10 (Side)

ED=AC=13 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

Answer

Angles BAC is equal to angle DEF.

Exercise #2

Given: ΔABC isosceles

and the line AD cuts the side BC.

Are ΔADC and ΔADB congruent?

And if so, according to which congruence theorem?

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Step-by-Step Solution

As we know the triangle is isosceles, then AC=AB

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

Given that the line AD intersects side BC, and therefore BD=DC

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

Answer

Congruent by L.L.L.

Exercise #3

Look at the triangles in the diagram.

Determine which of the statements is correct.

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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, then the angle ACB is 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.

Answer

All of the above.

Exercise #4

The triangles ABO and CBO are congruent.

Which side is equal to BC?

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Video Solution

Step-by-Step Solution

Let's consider the corresponding congruent triangles letters:

CBO=ABO CBO=ABO

That is, from this we can determine:

CB=AB CB=AB

BO=BO BO=BO

CO=AO CO=AO

Answer

Side AB

Exercise #5

Look at the triangles in the diagram.

Which of the following statements is true?

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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 the angle between them, 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.

Answer

It is not possible to calculate.

Exercise #1

Triangles ABC and CDA are congruent.

Which angle is equal to angle BAC?

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Video Solution

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 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).

Answer

C

Exercise #2

Are the triangles in the image congruent?

If so, according to which theorem?

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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.

Answer

No.

Exercise #3

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

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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.

Answer

Angle ABC equals 65.

Exercise #4

Which of the triangles are congruent?

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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.

Answer

It is not possible to know based on the data.

Exercise #5

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

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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.

 

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.

 

Answer

Congruent according to S.A.S.

Exercise #1

Are the triangles in the drawing congruent?

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Step-by-Step Solution

For triangles to be congruent, it is necessary to demonstrate that the S.A.S theorem is satisfied

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

Now we will look for the lengths of the other sides:

2X+4=X+2 2X+4=X+2

We proceed with the sections accordingly:24=2XX 2-4=2X-X

2=X -2=X

We place it in the right triangle and will find the length of the side:2+2=0 -2+2=0

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

Answer

No

Exercise #2

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

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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.

Answer

Data cannot be added for the triangles to be congruent.

Exercise #3

AD = BC

AD || BC

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

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Step-by-Step Solution

We are given: AD=BC

The angle ADB is equal to the angle CBD since AD is parallel to BC and the corresponding angles are equal between parallel lines.

DB=DB since it is a common side.

Therefore, we have two triangles that are congruent according to the S.A.S. (side, angle, side) theorem.

Answer

According to the S.A.S. theorem

Exercise #4

EDC is an isosceles triangle.

ADE=BCE ∢ADE=∢BCE

AC=BD AC=BD

ΔADEΔBCE ΔADE≅ΔBCE

According to which theorem are the triangles congruent?

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Step-by-Step Solution

ΔEDC is an isosceles triangle.

DE = EC

D=C ∢D=∢C

EDC=ECD ∢\text{EDC}=∢\text{ECD}

ADE=BCE ∢\text{ADE}=∢\text{BCE} (A)

E1=E2 ∢E1=∢E2 (A)

Reasoning:

In an isosceles triangle there are 2 equal sides.

The base angles of an isosceles triangle are equal.

DEDC=CECD ∢D-∢\text{EDC}=∢C-∢ECD

Therefore, the triangles are congruent according to the theorem S.A.S. theorem.

Answer

A.S.A.

Exercise #5

Look at the triangles in the diagram.

Which of the statements is true?

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Video Solution

Answer

Angle E is equal to angle B.