Congruence Of Triangles Class 7 Maths Notes

Key Definitions

Congruence

Two geometric figures are congruent if they have exactly the same shape and the same size. They can be perfectly superimposed on each other.

Congruent Figures

Figures that are exact copies of each other. The symbol for congruence is '≅'.

Corresponding Parts

When two figures are congruent, their matching sides, angles, and vertices are called corresponding parts. They occupy the same relative position in both figures.

Corresponding Vertices

Vertices that match up when two congruent figures are placed one over the other. The order of vertices in a congruence statement is crucial.

Corresponding Sides

Sides that match up when two congruent figures are placed one over the other. Their lengths are equal.

Corresponding Angles

Angles that match up when two congruent figures are placed one over the other. Their measures are equal.

CPCTC

An acronym for 'Corresponding Parts of Congruent Triangles are Congruent'. This principle states that if two triangles are proven congruent, then all their corresponding parts (sides and angles) are also equal.

Understanding Congruence

In geometry, congruence means being identical in form. If two figures are congruent, they are exact copies of each other – they have the same shape and the same size. Imagine taking one figure and placing it exactly on top of another; if they match up perfectly without any part sticking out or being covered, they are congruent. This process is called superimposition.

When we talk about congruence of triangles, it implies that if triangle ABC is congruent to triangle PQR (written as ΔABC ≅ ΔPQR), then:

• Corresponding vertices match: A corresponds to P, B to Q, and C to R.
• Corresponding sides are equal in length: AB = PQ, BC = QR, and CA = RP.
• Corresponding angles are equal in measure: ∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R.

It's very important to write the correspondence correctly. For example, if ΔABC ≅ ΔPQR, it implies a specific mapping of vertices. Writing ΔABC ≅ ΔQPR would mean A corresponds to Q, B to P, and C to R, which might not be true. The order of letters in the congruence statement directly indicates the corresponding vertices, sides, and angles. Understanding this correspondence is key to correctly applying congruence criteria and using the CPCTC property to find unknown sides or angles.

Criteria for Congruence of Triangles

To prove that two triangles are congruent, we don't always need to check if all three sides and all three angles are equal. There are specific minimum conditions, known as congruence criteria or congruence rules, that are sufficient to prove congruence:

1. SSS (Side-Side-Side) Congruence Criterion:

• Rule: If three sides of one triangle are equal to the three corresponding sides of another triangle, then the two triangles are congruent.
• Example: If in ΔABC and ΔPQR, AB = PQ, BC = QR, and CA = RP, then ΔABC ≅ ΔPQR.

1. SAS (Side-Angle-Side) Congruence Criterion:

• Rule: If two sides and the included angle (the angle between the two sides) of one triangle are equal to the two corresponding sides and the included angle of another triangle, then the two triangles are congruent.
• Crucial Point: The angle must be the one formed by the two given sides. If the angle is not included, SAS cannot be applied.
• Example: If in ΔABC and ΔPQR, AB = PQ, ∠B = ∠Q, and BC = QR, then ΔABC ≅ ΔPQR.

1. ASA (Angle-Side-Angle) Congruence Criterion:

• Rule: If two angles and the included side (the side between the two angles) of one triangle are equal to the two corresponding angles and the included side of another triangle, then the two triangles are congruent.
• Crucial Point: The side must be the one that connects the vertices of the two given angles.
• Example: If in ΔABC and ΔPQR, ∠B = ∠Q, BC = QR, and ∠C = ∠R, then ΔABC ≅ ΔPQR.

1. RHS (Right-angle-Hypotenuse-Side) Congruence Criterion:

• Rule: This criterion applies specifically to right-angled triangles. 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 two triangles are congruent.
• Components: Requires a Right angle (90°), the Hypotenuse (side opposite the right angle), and one other Side.
• Example: If in right-angled ΔABC (at B) and ΔPQR (at Q), AC = PR (hypotenuse) and BC = QR (a side), then ΔABC ≅ ΔPQR.

Illustrative Examples

• {"title":"Example 1 (SSS)","bodyMarkdown":"Q: Given two triangles ΔABC and ΔDEF where AB = DE = 5 cm, BC = EF = 7 cm, and CA = FD = 6 cm. Are the triangles congruent?\nA: Yes, since all three corresponding sides are equal (AB=DE, BC=EF, CA=FD), by SSS congruence criterion, ΔABC ≅ ΔDEF."}
• {"title":"Example 2 (SAS)","bodyMarkdown":"Q: In ΔPQR and ΔXYZ, PQ = XY = 4 cm, ∠P = ∠X = 60°, and PR = XZ = 5 cm. Are they congruent?\nA: Yes. We have two sides (PQ, PR) and the included angle (∠P) of ΔPQR equal to two corresponding sides (XY, XZ) and the included angle (∠X) of ΔXYZ. So, by SAS congruence criterion, ΔPQR ≅ ΔXYZ."}
• {"title":"Example 3 (ASA)","bodyMarkdown":"Q: Consider ΔLMN and ΔSTU. If ∠L = ∠S = 40°, LM = ST = 8 cm, and ∠M = ∠T = 70°. Are they congruent?\nA: Yes. Two angles (∠L, ∠M) and the included side (LM) of ΔLMN are equal to two corresponding angles (∠S, ∠T) and the included side (ST) of ΔSTU. By ASA congruence criterion, ΔLMN ≅ ΔSTU."}
• {"title":"Example 4 (RHS)","bodyMarkdown":"Q: ΔABC is right-angled at B, and ΔDEF is right-angled at E. If AC = DF = 10 cm (hypotenuse) and BC = EF = 6 cm (side). Are they congruent?\nA: Yes. Both are right-angled triangles. Their hypotenuses are equal (AC=DF), and one pair of corresponding sides is equal (BC=EF). By RHS congruence criterion, ΔABC ≅ ΔDEF."}

Key Points to Remember

• Congruence means exactly the same shape and size. The symbol is '≅'.
• For two figures to be congruent, they must superimpose perfectly.
• The order of vertices in a congruence statement (e.g., ΔABC ≅ ΔPQR) defines the correspondence of parts.
• CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is a fundamental result used after proving congruence.
• There are four main congruence criteria for triangles: SSS, SAS, ASA, and RHS.
• For SAS, the angle must be included (between the two sides).
• For ASA, the side must be included (between the two angles).
• RHS criterion applies only to right-angled triangles and requires the hypotenuse and one side.
• AAA (Angle-Angle-Angle) is NOT a congruence criterion; it proves similarity, not congruence.
• SSA (Side-Side-Angle) is NOT a congruence criterion, except in special cases like RHS.

Exam Tip: Avoiding Common Traps

When solving problems on congruence, always clearly state the congruence criterion you are using (SSS, SAS, ASA, RHS). Pay very close attention to the correspondence of vertices; writing ΔABC ≅ ΔPQR implies a specific mapping (A↔P, B↔Q, C↔R). Incorrect correspondence will lead to wrong conclusions, even if the triangles are indeed congruent. Double-check if the angle in SAS is truly the included angle and if the side in ASA is truly the included side. For RHS, ensure it's a right-angled triangle, and you have both the hypotenuse and a side.

Practice Questions with Solutions

• Q: If ΔXYZ ≅ ΔLMN, what side in ΔLMN corresponds to side XZ in ΔXYZ? A: Side LN.
• Q: Can two triangles be congruent if only their three angles are equal (AAA criterion)? A: No, AAA is not a congruence criterion. It proves similarity, not congruence.
• Q: What does CPCTC stand for and when is it used? A: CPCTC stands for 'Corresponding Parts of Congruent Triangles are Congruent'. It is used after two triangles have been proven congruent to deduce equality of their corresponding sides and angles.
• Q: Which congruence criterion requires a right angle? A: The RHS (Right-angle-Hypotenuse-Side) congruence criterion.

Frequently Asked Questions