Class 7 — Mathematics (NCERT)
Congruence of Triangles
Class 7
- ✓Understand the meaning of congruence for geometric figures.
- ✓Identify congruent line segments and angles.
- ✓Define congruence of triangles and identify corresponding parts (CPCTC).
- ✓Apply the SSS, SAS, ASA, and RHS congruence criteria to determine if two triangles are congruent.
- ✓Solve problems involving congruence of triangles.
Key concepts
Two geometric figures are said to be congruent if they have exactly the same shape and the same size. They are exact copies of each other. If we place one figure over the other, they should perfectly superimpose each other. The symbol for congruence is '≅'.
Two line segments are congruent if and only if they have the same length. For example, if line segment AB has length 5 cm and line segment CD has length 5 cm, then AB ≅ CD.
Two angles are congruent if and only if they have the same measure. For example, if ∠ABC measures 60° and ∠PQR measures 60°, then ∠ABC ≅ ∠PQR.
Two triangles are congruent if and only if their corresponding parts (sides and angles) are equal. This means that if ΔABC ≅ ΔPQR, then their corresponding vertices, sides, and angles are equal. This is often referred to as CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
If three sides of one triangle are respectively equal to the three corresponding sides of another triangle, then the two triangles are congruent. For example, in ΔABC and ΔPQR, if AB = PQ, BC = QR, and CA = RP, then ΔABC ≅ ΔPQR (by SSS criterion).
If two sides and the included angle (the angle between the two sides) of one triangle are respectively equal to the two corresponding sides and the included angle of another triangle, then the two triangles are congruent. For example, in ΔABC and ΔPQR, if AB = PQ, ∠B = ∠Q, and BC = QR, then ΔABC ≅ ΔPQR (by SAS criterion).
If two angles and the included side (the side between the two angles) of one triangle are respectively equal to the two corresponding angles and the included side of another triangle, then the two triangles are congruent. For example, in ΔABC and ΔPQR, if ∠B = ∠Q, BC = QR, and ∠C = ∠R, then ΔABC ≅ ΔPQR (by ASA criterion).
If in two right-angled triangles, the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one corresponding side of the other triangle, then the two triangles are congruent. For example, in right-angled ΔABC (at B) and right-angled ΔPQR (at Q), if AC = PR (hypotenuses) and AB = PQ (one side), then ΔABC ≅ ΔPQR (by RHS criterion).
Key facts to remember
- 1Congruent figures have the same shape and the same size.
- 2The symbol for congruence is '≅'.
- 3CPCTC stands for 'Corresponding Parts of Congruent Triangles are Congruent'.
- 4SSS criterion: If three sides of one triangle are equal to three corresponding sides of another triangle, the triangles are congruent.
- 5SAS criterion: If two sides and the included angle of one triangle are equal to two corresponding sides and the included angle of another triangle, the triangles are congruent.
- 6ASA criterion: If two angles and the included side of one triangle are equal to two corresponding angles and the included side of another triangle, the triangles are congruent.
- 7RHS criterion: For right-angled triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and one corresponding side of another triangle, the triangles are congruent.
Worked examples
Example 1
In the given figure, AD = CD and AB = CB. Show that ΔABD ≅ ΔCBD.
Answer
ΔABD ≅ ΔCBD
This example demonstrates the SSS criterion where all three sides of one triangle are equal to the corresponding three sides of the other triangle.
Example 2
In the given figure, OA = OB and OD = OC. Show that ΔAOD ≅ ΔBOC.
Answer
ΔAOD ≅ ΔBOC
The angle must be the 'included' angle, i.e., the angle formed between the two equal sides.
Example 3
In the given figure, line segment AB is parallel to line segment DC, and M is the midpoint of AD. Show that ΔAMB ≅ ΔDMC. Also, state if BM = CM.
Answer
ΔAMB ≅ ΔDMC, and yes, BM = CM.
Remember that when lines are parallel, alternate interior angles are equal. The side must be the 'included' side, i.e., the side between the two equal angles.
Example 4
In right-angled triangles ABC and PQR, if hypotenuse AC = hypotenuse PR and side BC = side QR, show that ΔABC ≅ ΔPQR. (Angles at B and Q are 90°).
Answer
ΔABC ≅ ΔPQR
The RHS criterion is specifically for right-angled triangles and requires the hypotenuse and one side to be equal.
Common mistakes
- ✗Confusing congruence with similarity (similar figures have the same shape but not necessarily the same size).
- ✗Not identifying corresponding vertices, sides, and angles correctly, leading to incorrect congruence statements (e.g., writing ΔABC ≅ ΔQPR instead of ΔABC ≅ ΔPQR).
- ✗Using 'SSA' (Side-Side-Angle) as a congruence criterion, which is not valid.
- ✗Not ensuring the angle in SAS is the *included* angle (between the two sides).
- ✗Not ensuring the side in ASA is the *included* side (between the two angles).
- ✗Applying the RHS criterion to triangles that are not right-angled.
Exam tips
- ★Always draw a clear diagram if one is not provided in the question.
- ★Carefully list the given information and what needs to be proved in a structured manner.
- ★Identify the two triangles you need to prove congruent and systematically list three pairs of equal corresponding parts.
- ★Clearly state the congruence criterion (SSS, SAS, ASA, or RHS) used to prove congruence.
- ★After proving triangles congruent, use CPCTC to prove other corresponding parts equal if required by the question.
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