# Congruence of Right Triangles (using the Pythagorean Theorem)

🏆Practice congruent triangles

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.

## Test yourself on congruent triangles!

Look at the triangles in the diagram.

Which of the statements is true?

Right triangle congruence takes into account the unique properties of right triangles and uses them to prove congruence.

We are already familiar with the usual congruence theorems:

Congruence according to Side-Angle-Side
Congruence according to Angle-Side-Angle
Congruence according to Side-Side-Side.

We will illustrate this with an example.

The graph shows two right triangles: $\triangle ABC$ and $\triangle DEF$.

Both triangles have a right angle (equal to $90^o$ degrees).

Moreover, in both triangles there is a perpendicular equal to $3$ (i.e., $AB = DE$), while the remaining one is equal to $5 (AC = DF)$.

If we were now to use the Pythagorean theorem, we would reach the size of the second perpendicular in each of the triangles and this perpendicular would come out equal to $4$, since it is the same calculation.

Therefore, we can always make use of the conclusion we have already reached, according to which when we are given two right triangles, in which one of them is perpendicular and the rest are equal to each other, respectively, we can conclude that these are congruent triangles.

If you are interested in this article you may also be interested in the following articles:

In Tutorela you will find a wide variety of mathematical articles

## Exercises on congruence of right triangles

### Exercise 1

The segments $AC$ and $BD$ intersect at the point$K$.

Given: the point $K$ intersects $BD$.

$AK=CK$

$AB⊥AC$

$DC⊥AC$

To which congruence theorem does $\triangle ABK\cong\triangle CDK$ belong ?

Solution

Since $AK=CK$

$AB$ is perpendicular to: $AC$

$\sphericalangle A=90°$

A perpendicular line makes a right angle of $90°$ degrees

$DC$ is perpendicular to: $AC$

$\sphericalangle C=90°$

A perpendicular line forms a right angle of $90°$ degrees.

$\sphericalangle C=\sphericalangle A=90°$

$BK=KD$

Since the point $K$ intersects $BD$

The overlapping triangles according to the theorem $S.A.S$

$S.A.S$

Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today

### Exercise 2

Given: quadrilateral $ABCD$ square.

And inside it contains the deltoid $KBPD$.

According to which congruence theorem do the triangles overlap $ΔBAK≅ΔBCP$?

Solution

$ABCD$ Square

$AB=BC$ In a square all sides are equal.

$KBPD$

Since it is a deltoid

$BK=BP$

In a deltoid two pairs of adjacent sides are equal.

$\sphericalangle C=\sphericalangle A$

The angles are equal in a square.

$\sphericalangle BPC=\sphericalangle BKA$

The angles of the deltoid $P$ and $K$ are equal.

Therefore $180-\sphericalangle K=180-\sphericalangle P$

$\sphericalangle ABK=\sphericalangle PBC$

If two angles are equal, so is the third angle.

Triangles are equal according to $S.A.S$

$S.A.S$

### Exercise 3

In the given drawing there is an isosceles triangle $\triangle ACB$

$AD$ is the median in the triangle $\triangle ACB$.

$∢\text{ADC}=∢\text{ADB}$

According to which congruence theorem do the triangles coincide $ΔADC≅ΔADB$?

Solution

Since $AD$ is the median

$CD=DB$

The median of the triangle goes from the vertex to the center of the opposite side and divides $CB$ in two at the point $D$

$\sphericalangle ADC=\sphericalangle ADB$

Given

$\sphericalangle ABD=\sphericalangle ACD$

Since the triangle $ACB$ is isosceles, in an isosceles triangle the base angles are equal.

The congruent triangles by the theorem of $A.S.A$

$A.S.A$

Do you know what the answer is?

### Exercise 4

In the given figure:

$AB=DC$

According to which congruence theorem do the triangles coincide $ΔADC≅ΔDAB$?

Solution

Given that $AB=DC$

$\sphericalangle BAD=\sphericalangle CDA$

Given that $\sphericalangle A_1=\sphericalangle D_1$

Given that $\sphericalangle A_2=\sphericalangle D_2$

Therefore:

$\sphericalangle A_1+\sphericalangle A_2=\sphericalangle D_1+\sphericalangle D_2$

$AD=AD$

Common side

The overlapping triangles according to the theorem $S.A.S$

Superposed according to $S.A.S$

### Exercise 5

The segments $AE$ and $BD$ are equal.

Given:

$∢\text{AFD}=∢\text{BFE}$

According to which congruence theorem do the triangles $ΔAEF≅ΔBDF$ coincide ?

Solution

$\sphericalangle BDF=\sphericalangle AEF=90°$

Given a right angle of $90°$ degrees

$AE=BD$

Given

$\sphericalangle AFE+\sphericalangle AFB=\sphericalangle BFA+\sphericalangle BFD$

Given that $\sphericalangle AFD=\sphericalangle BFE$

Therefore $\sphericalangle AFE=\sphericalangle BFD$

$\sphericalangle FAE=\sphericalangle FBD$

if in a triangle two angles are equal then the third angle is also equal

Congruent triangles according to $A.S.A$

Congruent according to $A.S.A$

## Review questions

What is a right triangle?

A right triangle is a figure that has three sides and has a right angle, that is, an angle of $90\degree$, like the one shown in the figure.

The $\triangle ABC$ is a right triangle.

What is right triangle congruence?

Recall that the congruence of figures refers when two figures have the same shape and their corresponding sides and angles are equal, in the case of right triangles, it must be exactly the same. The difference here is that right triangles already have a defining characteristic that identifies them. If we have two right triangles then we already know that one of its angles measures $90\degree$ and it is only a matter of seeing what congruence criteria is met to verify that they are congruent triangles.

What are the congruence criteria to determine if two right triangles are congruent?

There are four criteria to determine if two triangles are congruent or not, which are the following:

SAS- Side, Angle, Side: Two triangles are congruent when two of their sides and the angle between them measure the same.

ASA- Angle, Side, Angle: Two triangles are congruent when two of their corresponding angles and the side between them measure the same.

SSS- Side, Side, Side: Two triangles are congruent if their three corresponding sides measure the same.

SSA- Side, Side, Angle: Two triangles are congruent if two of their corresponding sides and an angle opposite one of them have the same measure.

When are two right triangles not congruent?

Two right triangles are not congruent when they do not meet any of the above congruence criteria, i.e., their corresponding sides and angles have different measures (they have different shapes).

Do you think you will be able to solve it?

## Examples with solutions for Congruence of Right Triangles (using the Pythagorean Theorem)

### 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

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.