Congruence of Triangles

🎯 What will you learn on this page?

👉 Understand congruent triangles and corresponding parts.

👉 Use SSS, SAS, ASA, AAS and RHS correctly.

👉 Write triangle correspondence in the correct order.

👉 Use CPCT after proving two triangles congruent.

👉 Understand why AAA and SSA are not general congruence rules.

👉 Solve simple proof-based and numerical problems.

👉 What Are Congruent Triangles?

Two triangles are congruent if they have exactly the same shape and size.

If one triangle can be placed over the other so that they match completely, the triangles are congruent.

The symbol for congruence is ≅.

For example:

△ABC ≅ △PQR

The order of the letters is important:

A ↔ P, B ↔ Q, C ↔ R

Therefore, the corresponding sides are

AB = PQ, BC = QR, CA = RP

and the corresponding angles are

∠A = ∠P, ∠B = ∠Q, ∠C = ∠R.

📚 What Is CPCT?

CPCT stands for Corresponding Parts of Congruent Triangles.

After two triangles have been proved congruent, their corresponding sides and angles can be stated as equal.

For example, if

△ABC ≅ △PQR,

then by CPCT,

AB = PQ, BC = QR, AC = PR

and

∠A = ∠P, ∠B = ∠Q, ∠C = ∠R.

🔹 Main Triangle Congruence Rules

Side-Side-Side (SSS)

If the three corresponding sides of two triangles are equal, the triangles are congruent.

⭐ How to use the SSS rule+

∵	In △ABC and △EFG
	AB = EF
	BC = FG
and	CA = GE
∴	△ABC ≅ △EFG (SSS)

Side-Angle-Side (SAS)

If two corresponding sides and the included angle between them are equal, the triangles are congruent.

Remember: in SAS, the equal angle must be the angle included between the two equal sides.

⭐ How to use the SAS rule+

∵	In △ABC and △EFG
	AB = EF
	∠BAC = ∠FEG
and	AC = EG
∴	△ABC ≅ △EFG (SAS)

Angle-Side-Angle (ASA)

If two corresponding angles and the included side between them are equal, the triangles are congruent.

⭐ How to use the ASA rule+

∵	In △ABC and △PQR
	∠A = ∠P
	AB = PQ
and	∠B = ∠Q
∴	△ABC ≅ △PQR (ASA)

🔹 Angle-Angle-Side (AAS)

If two corresponding angles and one corresponding non-included side of two triangles are equal, the triangles are congruent by AAS.

For example, if

∠A = ∠P,∠B = ∠Q, and AC = PR,

then

△ABC ≅ △PQR (AAS).

This also makes sense because when two angles of a triangle are known, the third angle is fixed by the angle-sum property.

Right angle-Hypotenuse-Side (RHS)

If two right triangles have equal hypotenuses and one pair of corresponding sides equal, the triangles are congruent.

⭐ How to use the RHS rule+

∵	In △ABC and △EFG
	∠A = ∠E = 90°
	BC = FG (hypotenuse)
and	AC = EG
∴	△ABC ≅ △EFG (RHS)

⚠️ Which Information Is Not Enough?

🔸 AAA Is Not a Congruence Rule

If all three corresponding angles of two triangles are equal, the triangles can have the same shape but different sizes.

Therefore, AAA generally proves similarity, not congruence.

For example, two equilateral triangles may have side lengths 3 cm and 6 cm. All their angles are 60°, but the triangles are not congruent.

🔸 SSA Is Not Generally a Congruence Rule

Knowing two sides and an angle that is not included between them does not always determine a unique triangle.

Therefore, SSA is not a general congruence criterion.

Do not confuse RHS with SSA. RHS is a special rule for right triangles, where the right angle is already fixed.

✏️ Solved Examples

✍️ Example 1: Identify the Correct Rule

In △ABC and △PQR,

AB = PQ, BC = QR and CA = RP.

All three corresponding sides are equal.

Therefore,

△ABC ≅ △PQR (SSS).

✍️ Example 2: Use SAS

In △ABC and △DEF,

AB = DE,AC = DF, and ∠A = ∠D.

The equal angle is included between the two equal sides.

Therefore,

△ABC ≅ △DEF (SAS).

✍️ Example 3: Use Correspondence and CPCT

If △ABC ≅ △PQR and BC = 7 cm, find QR.

From the order of the congruence statement,

B ↔ Q and C ↔ R.

So BC ↔ QR.

By CPCT,

QR = BC = 7 cm.

✍️ Example 4: Use RHS

Two right triangles have hypotenuses of length 10 cm and one pair of corresponding sides of length 6 cm.

Both triangles have a right angle, their hypotenuses are equal, and one corresponding side is equal.

Therefore, the triangles are congruent by RHS.

✍️ Example 5: Is AAA Enough?

Two triangles have angles 40°, 60°, 80° and 40°, 60°, 80°.

Are they definitely congruent?

No. Equal angles do not guarantee equal side lengths.

Therefore, AAA does not prove congruence.

🧩 Practice Questions

✍️ 1. In △ABC and △PQR, AB = PQ, BC = QR and AC = PR. Which congruence rule applies?

✍️ 2. In △ABC and △DEF, AB = DE, AC = DF and ∠A = ∠D. Which rule applies?

✍️ 3. Two triangles have two pairs of equal angles and the side included between those angles is also equal. Which rule applies?

✍️ 4. Two right triangles have equal hypotenuses and one pair of corresponding sides equal. Which rule applies?

✍️ 5. If △ABC ≅ △PQR and AB = 5.4 cm, find PQ.

✍️ 6. If △XYZ ≅ △LMN, which angle corresponds to ∠Y?

✍️ 7. Do three pairs of equal corresponding angles prove that two triangles are congruent? Give a reason.

✍️ 8. Which of the following is not generally sufficient to prove triangle congruence: SSS, SAS, ASA, SSA, RHS?

✅ Answers to Practice Questions+

1. SSS

2. SAS

3. ASA

4. RHS

5. PQ = 5.4 cm

6. ∠Y = ∠M

7. No. AAA can give the same shape but different sizes.

8. SSA

⚠️ Common Misconceptions

🔹 Any equal angle can be used in SAS.
No. The equal angle must be the included angle between the two corresponding equal sides.

🔹 ASA and AAS give exactly the same information.
Both can prove congruence, but in ASA the given side lies between the two equal angles; in AAS it does not.

🔹 AAA proves that two triangles are congruent.
No. AAA generally tells us that the triangles have the same shape, not necessarily the same size.

🔹 RHS works for every triangle.
No. RHS applies only to right triangles.

🔹 The order in △ABC ≅ △PQR does not matter.
It matters. The order tells us which vertices, sides and angles correspond.

✅ Quick Revision

🔹 Congruent triangles have the same shape and the same size.
🔹 The congruence symbol is ≅.
🔹 SSS: three corresponding sides are equal.
🔹 SAS: two sides and the included angle are equal.
🔹 ASA: two angles and the included side are equal.
🔹 AAS: two angles and a non-included corresponding side are equal.
🔹 RHS: right angle + hypotenuse + one corresponding side.
🔹 After congruence is proved, CPCT is used to state that corresponding parts are equal.
🔹 AAA is not a congruence rule.
🔹 SSA is not generally a congruence rule.
🔹 Always keep the corresponding vertices in the correct order when writing a congruence statement.