Congruence of Right Triangles (using the Pythagorean Theorem)

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:

• The Pythagorean Theorem
• The application of the Pythagorean theorem in an orthohedron or cuboid.
• Congruent Triangles:
• Congruence Criterion: Side, Angle, Side
• Congruence criterion: Angle, Side, Angle
• Congruence criterion: Side, Side, Side, Side
• Area of a right triangle

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Task

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 $S.A.S$

Answer

$S.A.S$

Task

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 $S.A.S$

Answer

$S.A.S$

Task

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 $A.S.A$

Answer

$A.S.A$

Task

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 $S.A.S$

Answer

Superposed according to $S.A.S$

Task

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 $A.S.A$

Answer

Congruent according to $A.S.A$

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).