Criteria for Congruence of Triangles

In geometry, congruent triangles are triangles that are exactly the same – they have the same size and shape. This means all corresponding sides are equal in length, and all corresponding angles are equal in measure. Instead of proving all six corresponding parts (three sides and three angles) are equal, we can use congruence criteria (also known as postulates and theorems) to establish congruence with less information. These criteria are rules that provide sufficient conditions to prove that two triangles are congruent.

Formulae

These are the main congruence criteria:

• SAS (Side-Angle-Side): If two sides and the included angle (the angle between them) of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the triangles are congruent.
• ASA (Angle-Side-Angle): If two angles and the included side (the side between them) of one triangle are congruent to the corresponding two angles and included side of another triangle, then the triangles are congruent.
• AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to the corresponding two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
• SSS (Side-Side-Side): If all three sides of one triangle are congruent to the corresponding three sides of another triangle, then the triangles are congruent.
• RHS (Right-angle, Hypotenuse, Side): If the hypotenuse and one side of a right-angled triangle are congruent to the hypotenuse and the corresponding side of another right-angled triangle, then the triangles are congruent. (Also sometimes called HL: Hypotenuse-Leg).

Examples

Example-1 (SAS):

Consider two triangles, $\triangle ABC$ and $\triangle DEF$. If $AB \cong DE$, $\angle B \cong \angle E$, and $BC \cong EF$, then $\triangle ABC \cong \triangle DEF$ by SAS.

Example-2 (SSS):

Consider two triangles, $\triangle PQR$ and $\triangle STU$. If $PQ \cong ST$, $QR \cong TU$, and $RP \cong US$, then $\triangle PQR \cong \triangle STU$ by SSS.

Theorem with Proof

Theorem: SAS (Side-Angle-Side) Congruence Theorem

If two sides and the included angle of one triangle are congruent to the corresponding two sides and the included angle of another triangle, then the two triangles are congruent.

Proof:

1. Given: $\triangle ABC$ and $\triangle DEF$ such that $AB \cong DE$, $\angle B \cong \angle E$, and $BC \cong EF$.

2. Construction: Imagine placing $\triangle ABC$ on $\triangle DEF$ such that point $B$ coincides with $E$, and $BC$ lies along $EF$. Since $BC \cong EF$, point $C$ coincides with $F$.

3. Reasoning: Since $\angle B \cong \angle E$, and $BC$ is placed along $EF$, side $BA$ falls along $ED$. Also since $AB \cong DE$, point $A$ coincides with $D$.

4. Conclusion: Since points $A$, $B$, and $C$ coincide with $D$, $E$, and $F$ respectively, then $\triangle ABC \cong \triangle DEF$ by definition of congruence.

Common mistakes by students

• Incorrectly identifying the included angle: Students often mistakenly identify an angle that is not between the given sides as the included angle.
• Using SSA (Side-Side-Angle): SSA is NOT a valid congruence criterion. Knowing two sides and a non-included angle doesn’t always guarantee congruence (the ambiguous case).
• Misunderstanding RHS: Students sometimes forget that RHS applies only to right-angled triangles. They might try to apply it to non-right triangles.
• Forgetting to identify corresponding parts correctly: When applying criteria like ASA and AAS, it’s crucial to correctly match the corresponding angles and sides.

Real Life Application

These criteria are used extensively in engineering and construction. For instance, when designing bridges or buildings, engineers use congruence to ensure that different parts of a structure are identical, which provides strength and stability. They also use these ideas when creating architectural models and scale drawings.

Fun Fact

The ancient Greeks, particularly Thales of Miletus, are credited with the first proofs of geometric theorems, including some foundational ideas related to triangle congruence.

Recommended YouTube Videos for Deeper Understanding

Q.1 Which of the following sets of conditions is NOT sufficient to prove that two triangles are congruent?

Check Solution

Ans: C

The SSA condition does not guarantee congruence because two different triangles can be formed.

Q.2 In triangles $ABC$ and $PQR$, $AB = PQ$, $\angle B = \angle Q$, and $BC = QR$. Which congruence criterion can be used to prove $\triangle ABC \cong \triangle PQR$?

Check Solution

Ans: C

We have two sides and the included angle equal.

Q.3 Two right-angled triangles have the hypotenuse and one side equal. Which congruence criterion is applicable?

Check Solution

Ans: C

In a right-angled triangle, we have the hypotenuse (R) and one side (S) equal to corresponding parts in the other right-angled triangle.

Q.4 Given $\triangle DEF$ with $\angle D = 60^\circ$, $\angle E = 50^\circ$ and in $\triangle GHI$, $\angle G = 60^\circ$, $EF=HI$ and $\angle I = 70^\circ$. Can we prove $\triangle DEF \cong \triangle GHI$? If so, which congruence criterion?

Check Solution

Ans: C

$\angle F = 180^\circ – 60^\circ – 50^\circ = 70^\circ$. Thus, $\angle E = \angle H$, $EF = HI$ and $\angle F = \angle I$.

Q.5 If $AB=DE$, $BC=EF$ and $AC=DF$, what is the congruence criterion by which $\triangle ABC \cong \triangle DEF$?

Check Solution

Ans: B

All the three sides of one triangle are equal to the corresponding three sides of the other triangle.

Next Topic: Properties of Isosceles Triangles