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.

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Test yourself on congruent triangles!

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Look at the triangles in the diagram.

Which of the statements is true?

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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.

1 - Congruence of right triangles

We will illustrate this with an example.

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

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

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


Exercises on congruence of right triangles

Exercise 1

Task

The segments AC AC and BD BD intersect at the pointK K .

Given: the point K K intersects BD BD .

AK=CK AK=CK

ABAC AB⊥AC

DCAC DC⊥AC

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

Given the point K intersects BD

Solution

Since AK=CK AK=CK

AB AB is perpendicular to: AC AC

A=90° \sphericalangle A=90°

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

DC DC is perpendicular to: AC AC

C=90° \sphericalangle C=90°

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

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

BK=KD BK=KD

Since the point K K intersects BD BD

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

Answer

S.A.S S.A.S


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Exercise 2

Task

Given: quadrilateral ABCD ABCD square.

And inside it contains the deltoid KBPD KBPD .

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

Exercise 2- Given a square quadrilateral ABCD square

Solution

ABCD ABCD Square

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

KBPD KBPD

Since it is a deltoid

BK=BP BK=BP

In a deltoid two pairs of adjacent sides are equal.

C=A \sphericalangle C=\sphericalangle A

The angles are equal in a square.

BPC=BKA \sphericalangle BPC=\sphericalangle BKA

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

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

ABK=PBC \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

Answer

S.A.S S.A.S


Exercise 3

Task

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

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

ADC=ADB ∢\text{ADC}=∢\text{ADB}

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

Exercise 3 - In the given drawing, there is an isosceles triangle ACB.

Solution

Since AD AD is the median

CD=DB CD=DB

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

ADC=ADB \sphericalangle ADC=\sphericalangle ADB

Given

ABD=ACD \sphericalangle ABD=\sphericalangle ACD

Since the triangle ACB 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

Answer

A.S.A A.S.A


Do you know what the answer is?

Exercise 4

Task

In the given figure:

AB=DC AB=DC

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

Exercise 4 In the given figure AB=DC

Solution

Given that AB=DC AB=DC

BAD=CDA \sphericalangle BAD=\sphericalangle CDA

Given that A1=D1 \sphericalangle A_1=\sphericalangle D_1

Given that A2=D2 \sphericalangle A_2=\sphericalangle D_2

Therefore:

A1+A2=D1+D2 \sphericalangle A_1+\sphericalangle A_2=\sphericalangle D_1+\sphericalangle D_2

AD=AD AD=AD

Common side

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

Answer

Superposed according to S.A.S S.A.S


Exercise 5

Task

The segments AE AE and BD BD are equal.

Given:

AFD=BFE ∢\text{AFD}=∢\text{BFE}

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

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

Solution

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

Given a right angle of 90° 90° degrees

AE=BD AE=BD

Given

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

Given that AFD=BFE \sphericalangle AFD=\sphericalangle BFE

Therefore AFE=BFD \sphericalangle AFE=\sphericalangle BFD

FAE=FBD \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 A.S.A

Answer

Congruent according to A.S.A A.S.A


Check your understanding

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° 90\degree , like the one shown in the figure.

2.a - What is a right triangle?

The ABC \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° 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).


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