Congruence of Triangles: Concept

Congruence of triangles is a fundamental concept in geometry that deals with the comparison of two triangles. Two triangles are considered congruent if they are exactly the same shape and size. This means that all corresponding sides are equal in length, and all corresponding angles are equal in measure. In simpler terms, if you could perfectly place one triangle on top of the other, they would completely coincide.

Understanding congruence is crucial because it allows us to deduce information about unknown sides and angles in a triangle by relating them to known sides and angles in another congruent triangle. It also forms the basis for many geometric proofs and constructions.

Formulae

While there isn’t a specific formula for congruence itself, there are congruence postulates (rules) that help determine if two triangles are congruent. These are often referred to using acronyms:

• SSS (Side-Side-Side): If all three sides of one triangle are equal in length to the corresponding three sides of another triangle, then the triangles are congruent.
• SAS (Side-Angle-Side): If two sides and the included angle (the angle between those two sides) of one triangle are equal 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 those two angles) of one triangle are equal 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 equal to the corresponding two angles and non-included side of another triangle, then the triangles are congruent.
• RHS (Right-angle-Hypotenuse-Side): (For right-angled triangles only) If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, then the triangles are congruent.

Examples

Here are two examples to illustrate the concept of congruent triangles:

Example-1:

Consider two triangles, $ABC$ and $DEF$. If $AB = DE$, $BC = EF$, $AC = DF$, then by the SSS postulate, $\triangle ABC \cong \triangle DEF$ (triangle ABC is congruent to triangle DEF).

This means that not only are the sides equal, but $\angle A = \angle D$, $\angle B = \angle E$, and $\angle C = \angle F$.

Example-2:

Consider triangles $PQR$ and $STU$. If $PQ = ST$, $\angle Q = \angle T$, and $QR = TU$, then by the SAS postulate, $\triangle PQR \cong \triangle STU$.

This means that the triangles are identical in shape and size, and their other corresponding sides and angles are also equal.

Theorem with Proof

Theorem: If two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle, then the triangles are congruent (SAS Postulate).

Proof:

Let’s consider two triangles, $\triangle ABC$ and $\triangle DEF$.

Given: $AB = DE$, $AC = DF$, and $\angle BAC = \angle EDF$.

To Prove: $\triangle ABC \cong \triangle DEF$.

Proof:

1. Imagine placing $\triangle ABC$ on top of $\triangle DEF$ such that point A coincides with point D, and side AB lies along side DE. Since $AB = DE$, point B will coincide with point E.
2. Since $\angle BAC = \angle EDF$, side AC will lie along side DF. Also, since $AC = DF$, point C will coincide with point F.
3. Since points A, B, and C coincide with points D, E, and F respectively, the triangles $\triangle ABC$ and $\triangle DEF$ are exactly the same.
4. Therefore, $\triangle ABC \cong \triangle DEF$ by the definition of congruence.

Common mistakes by students

Students often make these mistakes when dealing with congruent triangles:

• Incorrectly applying congruence postulates: Misunderstanding which sides and angles must correspond when applying SSS, SAS, ASA, AAS, or RHS. For instance, incorrectly assuming SAS when the angle is not *included* between the two sides.
• Forgetting the order of vertices when stating congruence: When writing $\triangle ABC \cong \triangle DEF$, it is crucial that the corresponding vertices are in the same order. For instance, if $AB = DE$, then B must correspond to E, etc. Writing $\triangle ABC \cong \triangle FED$ implies a different correspondence of sides and angles.
• Assuming congruence without sufficient information: Attempting to prove congruence without enough information, such as only knowing two sides and a non-included angle (which is not a valid congruence criterion). This can lead to the SSA (Side-Side-Angle) case, which is not always sufficient for proving congruence.
• Confusion between congruence and similarity: Confusing the concepts of congruence (same size and shape) and similarity (same shape, but not necessarily the same size). These are related but distinct ideas.

Real Life Application

The concept of congruent triangles has many real-life applications:

• Construction and Architecture: Ensuring that structures are stable and symmetrical often relies on congruent shapes. For example, roof trusses are often designed using congruent triangles to distribute weight evenly.
• Engineering: Designing bridges, buildings, and other structures involves calculating the size and angles of components. Congruent shapes are often used to create strong and balanced designs.
• Navigation and Surveying: Triangulation, a method used in surveying and GPS, uses congruent or similar triangles to determine distances and locations.
• Art and Design: Artists and designers use congruent shapes to create symmetry and balance in their work.

Fun Fact

The symbol for congruence, $\cong$, is a combination of the symbol for equality (=) and the symbol for similarity (~), highlighting the fact that congruent figures are both equal and similar.

Recommended YouTube Videos for Deeper Understanding

Q.1 Which of the following statements best describes the meaning of congruent triangles?

Check Solution

Ans: D

Congruent triangles are identical in all respects.

Q.2 If $\triangle ABC \cong \triangle DEF$, which of the following is true?

Check Solution

Ans: D

Corresponding sides and angles are equal in congruent triangles.

Q.3 Two triangles are congruent. If one triangle has sides of length 3 cm, 4 cm, and 5 cm, what are the side lengths of the other triangle?

Check Solution

Ans: B

Congruent triangles have equal corresponding sides.

Q.4 Which of the following conditions is NOT sufficient to prove that two triangles are congruent?

Check Solution

Ans: D

SSA is not always a valid congruence criteria.

Q.5 If $\triangle PQR \cong \triangle STU$, and $PQ = 5$ cm, $QR = 7$ cm, and $\angle Q = 60^{\circ}$, then which of the following statements must be true?

Check Solution

Ans: C

Corresponding sides and angles are equal. $QR$ corresponds to $TU$.

Next Topic: Criteria for Congruence of Triangles