# Congruence Criterion: Side, Angle, Side

🏆Practice side, angle, side

We'll study the three main congruence criteria. This is the first one of them:

## Side, Angle, Side.

According to this theorem, two triangles are congruent if two of their sides are respectively equal and the angle between them is also equal.
It is important to note that the angle must be between the two equal sides. This criterion cannot be applied if it were a different angle.

To demonstrate that two triangles are congruent, we can use one of the following postulates:

• SAS - Side, Angle, Side
• ASA - Angle, Side, Angle
• SSS- Side, Side, Side
• HL- Hypotenuse, Leg

## Test yourself on side, angle, side!

AB = CD

$$∢\text{BAC}=∢\text{DCA}$$

According to which theorem are triangles Δ ABC and Δ CDA congruent?



## Definition of Congruent Triangles

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.

## Example 1 (Side, Angle, Side)

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

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## Example 2 - Congruence (Side, Angle, Side)

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.

## Side-Angle-Side Congruence Exercises

### Exercise 1

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

$ASA$ (Angle-Side-Angle)

Do you know what the answer is?

### Exercise 2

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)

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

### Exercise 3

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)

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

### Exercise 4

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.

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

### Exercise 5

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)

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

Do you think you will be able to solve it?

### Exercise 6

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.

According to the $ASA$ criterion.

## Review Questions

### 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.

## examples with solutions for side, angle, side

### Exercise #1

AB = CD

$∢\text{BAC}=∢\text{DCA}$

According to which theorem are triangles Δ ABC and Δ CDA congruent?