Congruence Criterion: Side, Angle, Side

Two triangles are congruent if two sides and the angle between them are equal in measure.

This criterion helps us prove that two angles are congruent.
Attention! The angle must be the one that is between the two equal sides. This theorem cannot be applied if it is another angle.

Given two triangles $Δ ABC$ and $Δ DEF$ and the following data:

$AB = DE$

$∠B=∠E$

$BC = FE$

From this, it can be deduced that the triangles $Δ ABC$ and $Δ DEF$ are congruent; therefore, we will write:

$Δ DEF ≅ Δ ABC$ according to the Side, Angle, Side (SAS) congruence criterion

On side $BD$, two triangles have been constructed: triangle $Δ ABD$ and triangle $ΔBCD$ such that:

$AD = DC$

$∢BDA = ∢BDC$

Prove that $∢BAD = ∢BCD$

Proof:

We will use the criterion that we have learned to prove that triangle $Δ ABD$ and triangle $ΔCBD$ are congruent triangles.

We note that side $BD$ is common to both triangles (edge)

It is also shown that: $∢BDA = ∢BDC$ (angle)

and that: $AD = DC$ (edge)

Consequently, we will deduce that $Δ CBD ≅ Δ ABD$ according to the Side, Angle, Side (SAS) congruence criterion.

It is crucial to pay attention and write the correct order of the vertices.

After seeing that the triangles are congruent, we can conclude that $∢BAD = ∢BCD$ (Corresponding angles in congruent triangles).

If you're interested in this article, you might also be interested in the following articles:

Congruence Criterion: Angle, Side, Angle

Congruence Criterion: Side, Side, Side

Side, Side and the Angle Opposite the Larger of the Two Sides

Style of Writing a Formal Proof in Geometry

On the Tutorela blog, you'll find a variety of articles about mathematics.

Given: $AM$ is the bisector of $∢\text{BMK}$

$∢\text{BMK}=100°$

$∢\text{KBM=50\degree}$

$∢A=∢K$

To which congruence theorem does $ΔABM≅ΔBKM$ belong?

Solution

$BM=BH$ Common side

Angle $A$ is equal to angle $K$ given

Angle $BMK$ is given as $100°$

Angle $KBM$ is given as $50°$

Angle $HMB$ is $50°$ because $AM$ is the bisector of $BMK$

Angle $KBM$ is equal to angle $AMB$, therefore both are $50°$

Angle $ABM$ is equal to angle

$KMB$

If two angles in a triangle are equal, then the third angle will also be equal, and the triangles will overlap according to the Angle-Side-Angle (ASA) congruence theorem

Answer

$ASA$ (Angle-Side-Angle)

Prompt

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

Given: Point $K$ intersects $BD$.

$AK=CK$

$AB⊥AC$

$CD⊥AC$

By what criterion of congruence are triangles $ΔABK≅ΔCDK$?

Solution

$AK=CK$

$AB$ is perpendicular to $AC$

A line perpendicular creates a right angle of $90^o$ degrees, therefore, angle $A$ is equal to:$90^o$ degrees

$CD$ is perpendicular to $AC$

A line perpendicular creates a right angle of $90^o$ degrees, therefore, angle $C$ is equal to: $90^o$ degrees

From this it follows that the angles

$A=C=90^o$

$BK=KD$

Given point $K$ which intersects $BD$

$BD$

Therefore the triangles are congruent according to the criterion$S.A.S$ (Side, Angle, Side)

Answer

Congruent: $S.A.S$ (Side, Angle, Side)

Assignment

In the given figure:

$AB=CD$

$∢BAC=∢DCA$

According to which criterion of congruence are $ΔABC≅ΔCDA$?

Solution

Given that

$AB=CD$

Given that the angles

$BAC=DCA$

Side $AC=AC$ is a common side

The triangles are congruent by the $SAS$ theorem (side, angle, side)

Answer

Congruent by $SAS$ (side, angle, side) criterion

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Task

Are the triangles in the drawing congruent?

If so, explain according to which criterion.

Solution

$AB=AB=4$

$AC=AC=12$

Angles $ACB=ACB=60º$

The triangles are congruent by the $S.A.S$ (side, angle, side) congruence criterion.

Answer

Congruent by $S.A.S$ (side, angle, side)

Prompt

Are the triangles $CDE$ and $ABE$ congruent?

If so, according to which congruence criterion?

Solution

Given that $AE=ED$

angles $BAE=EDC=50^\circ$

angle $E=E$ are vertically opposite angles

The triangles are congruent according to the critterion $AAS$ (angle, angle, side)

Answer

Congruent by $AAS$ (angle, angle, side)

Assignment

Given the figure:

$AD=BC$

$AD\parallel BC$

By which theorem do the triangles $\triangle ABD\cong \triangle BCD$ coincide?

Solution

Given that $AD=BC$

Given that $AD$ is parallel to $BC$

Alternate interior angles $\angle ADB=\angle CBD$ between parallel lines are equal

$BD=BD$ common side

Triangles are congruent according to the $ASA$ criterion.

Answer

According to the $ASA$ criterion.

What is a triangle?

In geometry, a triangle is considered a flat figure with three sides, where the joining of each side, called vertices, forms three angles.

What are congruent triangles?

If two triangles have sides and angles of the same measure, then they are congruent triangles.

What criteria can be used to determine if two triangles are congruent?

There are four criteria to determine whether two triangles are congruent or not, which are as follows:

• SAS - Side, Angle, Side.
• ASA - Angle, Side, Angle.
• SSS - Side, Side, Side.
• SSA - Side, Side, Angle.

What is the Side, Angle, Side criterion?

This criterion tells us that two triangles are congruent when two of their corresponding sides and the angle between them are equal. It should be noted that if the angle analyzed is not the one between these two sides, we cannot use this criterion.

For what types of triangles can we use the congruence criteria?

We can apply the criteria to any type of triangle, whether it's an equilateral triangle, an isosceles triangle, or a scalene triangle.