Congruent Triangles

In this chapter we will learn about the concept of congruent triangles and will also look at the condition which makes two triangles congruent.

What are Congruent Triangles?

Two triangles are congruent when on overlapping they get completely superimposed over each other.

Hence, the congruent triangles have exactly same shape, same side length and same angle measurement.

When everything is same in congruent triangles then why student find it difficult to identify same triangles?

Isn’t it easy to just look at the triangles and identify if they are congruent or not?

Identifying congruent triangles are difficult as ;

⟹ if triangles are pointed in different direction which makes it difficult to identify.

⟹ small difference in angles are difficult to identify.

In first look the triangle may appear the same but it need deep analysis to check if they have same angles or not.

Hence we need some shortcut method to identify the congruent triangles.

In this chapter we have discussed both the conventional and shortcut rules to speed up the identification process.

Representing congruency of Triangles

Congruent triangles are represented by symbol \mathtt{\cong }

Consider the below triangles ABC & DEF.

Both the triangles are congruent. It means that when we superimpose triangle ABC on DEF, they will fully overlap each other.

So we can write;
▵ABC \mathtt{\cong } ▵DEF

When triangles are congruent it means that they have equal sides and angles.

Showing equal sides in diagram

In above triangles ABC & DEF, the equal sides are represented by perpendicular lines on corresponding sides.

Single red line on sides AB & DE signifies that both sides are equal.

AB = DE

Double Blue line on BC & EF signify that they are equal in length.

BC = EF

Triple Green line on AC & DF signifies that they are of equal length.

AC = DF

Showing equal angles in Diagram

In triangles ABC & DEF, equal angles are represented by circles around the angles.

∠A & ∠D have single red mark around them. The mark tells that both angles are equal.
∠A = ∠D

∠B & ∠E have double green mark around the angle.
∠B = ∠E

∠C & ∠F have triple blue mark.
∠C = ∠F

Conclusion
Congruent triangles have equal sides and angles.
They can perfectly superimpose over each other.

Congruent triangles rule

While checking the congruency of triangles, finding equal sides and angles can take up lot of time.

Instead of checking each sides and angles, follow the below rules for proving congruency between triangles.

(a) SSS congruency

SSS stands for side – side – side congruency.

It states that triangles are congruent of three sides of triangle are equal to the corresponding sides of other triangle.

Hence, if sides are equal there is no need to check the angles of triangle. Three equal sides implies that the angles are also equal.

Consider the above triangle ABC & PQR

Here;
AB = QR = 4 cm
AC = PR = 3.44 cm
BC = PQ = 5.62 cm

Since all three sides are equal, ▵ABC is congruent to ▵ QRP.

i.e. ▵ ABC \mathtt{\cong } ▵QRP

(b) SAS congruency

SAS stands for side- angle – side.

The rule states for given triangles if two sides and angles between them are equal then the triangles are congruent.

In the above triangles ▵ABC and ▵PQR, two sides and angle between them are equal.

AB = QR
∠B = ∠Q
BC = PQ

Hence by SAS congruency, both the triangles are congruent.

i.e. ▵ABC \mathtt{\cong } ▵RQP

Note:
The equal angle in SAS should lie between the equal side otherwise this rule will not be applicable.

(c) ASA congruency

ASA stands for angle – side – angle.

Triangles are congruent when the two angles and the side between them are equal.

In the above triangles ABC and PQR, two angles and side between them are equal.

∠B = ∠Q
AB = PQ
∠A = ∠R

Hence by ASA congruency, both the triangles are congruent.

i.e. ▵ABC \mathtt{\cong } ▵RQP

Note:
The equal side should lie between the equal pair of angles otherwise this rule will not be valid.

(d) AAS congruency

AAS stands for Angle – Angle – Side

Two triangles are congruent if two angles and non included side are equal.

Non included side is the one which doesn’t lie between the equal angles.

In the above triangle;
∠B = ∠Q
∠A = ∠R
AC = PR

Hence by AAS congruency, both the triangles are congruent.

i.e. ▵ABC \mathtt{\cong } ▵RQP

Note:
The equal side should not placed in between the equal angles otherwise this rule will not be applicable.

(e) RHS congruency

RHS stands for Right angle – Hypotenuse – Side.

This rule states for for any given right triangles if the hypotenuse and other sides of triangles are equal then the triangles will be congruent.

In the above triangles ABC & PQR

∠B = ∠R { Right angle }
AC = PQ { hypotenuse}
BC = QR { Side }

Using RHS congruency we can say that both triangles are congruent.

i.e. ▵ABC \mathtt{\cong } ▵PRQ

Note:
This rule is only applicable for right angle triangle.

Next chapter : Understand SSS congruency condition in detail