Chapter 7 — Congruence Of Triangles

Congruence means that two geometric figures have exactly the same shape and size. If one figure can be moved (translated, rotated or reflected) so that it fits exactly on the other, the two figures are called congruent. The symbol for congruence is ≅ (for example, ΔABC ≅ ΔDEF).

For triangles, congruence means that their corresponding sides are equal in length and corresponding angles are equal in measure. Corresponding parts are matched in order: if ΔABC ≅ ΔDEF then AB corresponds to DE, BC to EF, CA to FD and ∠A to ∠D, ∠B to ∠E, ∠C to ∠F.

Key ideas used to check congruence (called congruence criteria) are tests that compare a few corresponding parts rather than all six parts. Common criteria used in school geometry are SSS (three sides equal), SAS (two sides and the included angle equal), ASA (two angles and the included side equal) and RHS (right-angle, hypotenuse and one side equal for right triangles). If a congruence criterion is satisfied, the two triangles are congruent and all other corresponding sides and angles are equal.

Note the difference from similarity: similar figures have the same shape but not necessarily the same size (their corresponding sides are proportional), while congruent figures have both shape and size identical.

What is congruence? Two shapes are congruent if they have exactly the same shape and size. For triangles, congruence means all corresponding sides are equal in length and all corresponding angles are equal in measure.

How to write congruence of triangles: If triangle ABC is congruent to triangle PQR, we write ΔABC ≅ ΔPQR. The order of letters shows the correspondence: A ↔ P, B ↔ Q, C ↔ R.

What congruence implies:

• Corresponding sides are equal (AB = PQ, BC = QR, CA = RP).
• Corresponding angles are equal (∠A = ∠P, ∠B = ∠Q, ∠C = ∠R).
• Perimeters and areas of congruent triangles are equal.

Common congruence criteria (how to prove two triangles congruent):

• SSS (Side–Side–Side): If three sides of one triangle are respectively equal to three sides of another triangle, the triangles are congruent.
• SAS (Side–Angle–Side): If two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle, the triangles are congruent.
• ASA (Angle–Side–Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
• AAS (Angle–Angle–Side): If two angles and a non-included corresponding side of one triangle are equal to those of another triangle, the triangles are congruent.
• RHS (Right angle–Hypotenuse–Side): For right triangles, if the hypotenuse and one side of one right triangle are equal to the hypotenuse and corresponding side of another, they are congruent.

Important caution: SSA (two sides and a non-included angle) is not generally a valid test for congruence because it can give two different triangles (ambiguous case).

How to use correspondence and notation: Always name triangles so corresponding vertices match. For example, if you prove AB = DE, BC = EF and ∠B = ∠E and you wish to conclude congruence, write the triangles in corresponding order: ΔABC ≅ ΔDEF (A↔D, B↔E, C↔F).

Quick strategy to prove congruence:

• Mark equal sides with identical tick marks and equal angles with identical arcs.
• Decide which congruence criterion applies (SSS, SAS, ASA, AAS, or RHS).
• Use the chosen criterion to state the congruence and then deduce equal corresponding parts (angles/sides).

What is congruence? Two figures are congruent when they have the same shape and size. For triangles, congruence means their corresponding sides and corresponding angles are equal. We write, for example, ΔABC ≅ ΔDEF to mean triangle ABC is congruent to triangle DEF.

Corresponding parts: When two triangles are congruent, each vertex, side and angle of one triangle matches a specific vertex, side and angle of the other. Correspondence must be consistent: the order of letters in the notation shows which vertices correspond (A ↔ D, B ↔ E, C ↔ F in ΔABC ≅ ΔDEF).

Why we need criteria: To prove two triangles are congruent we do not need to check all six parts. Certain combinations of equal parts are enough. These are the standard criteria taught in Class 7.

• SSS (Side–Side–Side): If the three sides of one triangle are respectively equal to the three sides of another triangle, the triangles are congruent. (AB = DE, BC = EF, CA = FD implies ΔABC ≅ ΔDEF.)
• SAS (Side–Angle–Side): If two sides and the included angle (the angle between the two sides) of one triangle are respectively equal to two sides and the included angle of another triangle, the triangles are congruent. (AB = DE, ∠B = ∠E, BC = EF implies ΔABC ≅ ΔDEF.)
• ASA (Angle–Side–Angle): If two angles and the included side (the side between those two angles) of one triangle are respectively equal to two angles and the included side of another triangle, the triangles are congruent. (∠A = ∠D, AB = DE, ∠B = ∠E implies ΔABC ≅ ΔDEF.)
• RHS (Right angle–Hypotenuse–Side): For right-angled triangles, if the hypotenuse and one other side of one right triangle are equal to the hypotenuse and corresponding side of another right triangle, the triangles are congruent. (For right triangles ΔABC and ΔDEF with right angles at B and E: hypotenuse AC = DF and side AB = DE implies congruence.)

Important note: SSA (Side–Side–Angle) is not a valid criterion in general because it may give two different (non-congruent) triangles — the ambiguous case. Also, AAS (Angle–Angle–Side) is effectively covered because two angles determine the third, so AAS implies ASA.

How to use the criteria in proofs: 1) Identify corresponding vertices. 2) Mark the equal parts given. 3) Choose the correct criterion (SSS, SAS, ASA, or RHS). 4) Conclude congruence and then state equalities of other corresponding parts (CPCTC: Corresponding Parts of Congruent Triangles are Congruent).

Introduction: In geometry, two figures are called congruent when they have the same shape and size. For triangles, congruence means every corresponding side and every corresponding angle of one triangle is equal to the corresponding side or angle of the other.

Basic definitions:

• Triangle (Δ): A polygon with three sides and three angles. Vertices are usually named A, B, C.
• Vertex: A corner point of a triangle (for example, A, B, C).
• Side: A line segment joining two vertices (for example, AB, BC, CA).
• Angle: The inclination between two sides at a vertex (for example, ∠A, ∠B, ∠C).
• Corresponding parts: When two triangles are congruent, matching vertices, sides and angles are corresponding (e.g., if ΔABC ≅ ΔPQR then A↔P, B↔Q, C↔R).
• Included angle: The angle formed between two given sides. For sides AB and AC, the included angle is ∠A.
• Included side: The side between two given angles. For angles ∠A and ∠B, the included side is AB.
• Hypotenuse: The longest side opposite the right angle in a right triangle (used in right-triangle congruence criteria).

Notation and correspondence: We write ΔABC ≅ ΔPQR to mean triangle ABC is congruent to triangle PQR. The order matters: vertex A corresponds to P, B to Q, C to R. From this we infer AB = PQ, BC = QR, CA = RP and ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R.

Common congruence criteria (summary):

• SSS (Side–Side–Side): If three sides of one triangle are respectively equal to three sides of another, the triangles are congruent.
• SAS (Side–Angle–Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another, the triangles are congruent.
• ASA (Angle–Side–Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another, the triangles are congruent.
• RHS (Right angle–Hypotenuse–Side): For right triangles: if the hypotenuse and one other side of a right triangle are equal to the hypotenuse and one other side of another right triangle, the triangles are congruent.

How to check congruence in practice: Use tick marks on equal sides and small arcs for equal angles; ensure the correspondence order is correct when applying a criterion (order of vertices is essential).

Why it matters: Congruence lets us assert equal lengths and angles without measuring everything—used in proofs, constructions, engineering and design.

What it means: Using congruence to prove results means we show two triangles are congruent (i.e. identical in shape and size) by applying a congruence criterion, and then use the fact that corresponding parts of congruent triangles are equal (CPCTC) to deduce the required result.

General steps:

1. Identify two triangles that are relevant to the statement to be proved.
2. Show pairs of corresponding sides/angles are equal so that one congruence criterion applies (SSS, SAS, ASA, AAS or RHS).
3. Conclude the two triangles are congruent.
4. Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to obtain the desired conclusion (e.g. equality of some sides or angles).

Why it works: Congruence gives a one-to-one correspondence between elements (vertices, sides, angles) of the two triangles. Once congruence is established, any corresponding side or angle in one triangle equals the other, so properties can be proved rigorously.

What are congruent triangles? Two triangles are congruent if all their corresponding sides and corresponding angles are equal. We write ΔABC ≅ ΔA'B'C' when AB = A'B', BC = B'C' and CA = C'A', and corresponding angles are equal.

Tools used: ruler (for straight lines) and compass (for copying lengths and drawing arcs).

Congruence criteria (used for construction): SSS (Side–Side–Side), SAS (Side–Angle–Side), ASA (Angle–Side–Angle), and RHS (Right angle–Hypotenuse–Side) for right triangles. These tell us which data uniquely determine a triangle and so allow construction of a triangle congruent to a given one.

1. Construction by SSS (given three sides): Given ΔABC (lengths AB, BC, CA). To construct ΔA'B'C' congruent to it:
   1. Draw a line and mark segment A'B' = AB.
   2. With center A' draw an arc of radius A'C' = AC.
   3. With center B' draw an arc of radius B'C' = BC.
   4. The intersection of the two arcs is C'. Join A'C' and B'C' to complete ΔA'B'C'.
2. Construction by SAS (two sides and included angle): Given ΔABC with AB, AC and included ∠A. To build ΔA'B'C':
   1. Draw A'B' = AB.
   2. At A' construct an angle equal to ∠A (use compass/angle copying).
   3. On one ray of that angle mark A'C' = AC.
   4. Join C' to B' to form ΔA'B'C'.
3. Construction by ASA (two angles and included side): Given ΔABC with ∠A, ∠B and side AB. To construct ΔA'B'C':
   1. Draw A'B' = AB.
   2. At A' construct an angle equal to ∠A; at B' construct an angle equal to ∠B.
   3. The rays from A' and B' meet at C'. Join C' to A' and B' to complete the triangle.
4. Construction by RHS (right triangle: hypotenuse and one side): For right ΔABC with right angle at C, given hypotenuse AB and a leg AC:
   1. Draw A'B' = AB.
   2. Draw the circle with diameter A'B' (all points on it subtend a right angle).
   3. With center A' draw an arc of radius A'C' = AC. The intersection of this arc with the circle (on the correct side) is C'.
   4. Join C' to A' and B' to get the required triangle (right angle at C').

Important note: Not every set of three measurements determines a unique triangle (for example SSA is ambiguous). Always use recognized congruence criteria (SSS, SAS, ASA, RHS) for unique constructions.

Overview

In the topic "Problem Solving and Examples" under the Chapter "Congruence of Triangles" (Class 7), we apply congruence criteria to determine when two triangles are identical in shape and size. Two triangles are congruent if their corresponding sides and corresponding angles are equal. The standard congruence tests used to solve problems are SSS, SAS, ASA and RHS. Problems typically ask you to prove triangles congruent and then use congruence to find unknown sides or angles (using CPCTC — Corresponding Parts of Congruent Triangles are Equal).

General approach to solving problems

• Read the problem and mark all given data on the figure (side lengths, angles, right angles, parallel lines, etc.).
• Look for one of the congruence criteria (SSS, SAS, ASA, RHS) by identifying matching sides/angles.
• State the congruence (for example, ΔABC ≅ ΔDEF by SAS) and list the corresponding parts.
• Use CPCTC to find required unknowns (sides or angles).
• Write a clear, concise proof/solution with steps and reasons.

Tips

• Always clarify which parts correspond before using CPCTC.
• If given parallel lines, use alternate interior or corresponding angles to show equality of angles.
• Mark equal sides with single/double tick marks and equal angles with arc marks to keep track visually.

Real-life connection

Congruence is used in construction (ensuring identical parts), manufacturing (duplicate components), computer graphics (copying shapes without distortion), and pattern-making (ensuring symmetry and identical repeats).

What it is: Logical reasoning in geometry means using given facts and accepted rules to reach a correct conclusion. Proof writing is the clear, step-by-step presentation of that reasoning so others can check it. In the context of congruence of triangles, you use given equalities (sides/angles) and accepted congruence criteria to conclude two triangles are identical in shape and size.

Structure of a good geometric proof

• Given: List what is known (e.g., AB = DE, ∠A = ∠D).
• To Prove: State the goal (e.g., ΔABC ≅ ΔDEF).
• Construction (if needed): Draw extra lines or points required for the proof.
• Proof / Reasoning: Present clear statements with reasons (each step justified: definition, axiom, congruence criterion, or previously proved result).
• Conclusion: Summarise the final result (often using CPCTC — corresponding parts of congruent triangles are equal).

Common logical connectors and phrases: "Since", "Because", "Therefore", "Hence", "So", "It follows that", "From (given or earlier step)". Use them to show how one statement leads to the next.

Congruence idea used in reasoning: Two triangles are congruent if all corresponding sides and angles match according to accepted criteria (SSS, SAS, ASA, RHS). Once congruence is established, every corresponding side and angle are equal — a fact often used to prove further properties.

Tips for neat proofs:

• Label the figure clearly; mark equal sides with identical tick marks and equal angles with identical arcs.
• Refer to the exact parts (e.g., "side AB corresponds to DE").
• Write short, numbered statements and give a reason for each.
• If using a congruence criterion, explicitly mention which criterion (SSS, SAS, ASA, or RHS) you are applying.

What is congruence of triangles?

Two triangles are said to be congruent if they are exactly same in shape and size — that is, their corresponding sides are equal in length and corresponding angles are equal in measure. We use the symbol ≅ to write congruence: for example, ΔABC ≅ ΔDEF means side AB corresponds to DE, BC to EF, and CA to FD.

Important points and rules

• Corresponding parts: When two triangles are congruent, corresponding sides are equal and corresponding angles are equal. The order of letters in the congruence statement matters: ΔABC ≅ ΔDEF means A↔D, B↔E, C↔F.
• Congruence criteria (ways to prove triangles are congruent): The commonly used criteria in Class 7 are SSS, SAS, ASA and RHS (for right-angled triangles). These give a minimal set of matching elements to conclude congruence.
• CPCT (Corresponding Parts of Congruent Triangles): Once two triangles are proved congruent, any corresponding side or angle in one triangle is equal to the corresponding part in the other.
• Notation & Marks: Use single, double, triple tick-marks on sides to show equal lengths and arc marks on angles to show equality. Right angles are marked with a small square.
• Order of vertices: Always match vertices in the same correspondence while writing a congruence statement. Incorrect ordering changes the correspondence and may lead to wrong conclusions.

How to approach a congruence question

1. Identify the given equal sides/angles and mark them clearly on the diagram.
2. Decide which congruence criterion (SSS, SAS, ASA, or RHS) fits the given information.
3. State the congruence with correct vertex order, e.g., ΔPQR ≅ ΔXYZ.
4. Use CPCT to state any further required equalities (sides or angles).

Common mistakes to avoid

• Assuming triangles are congruent just because they look identical — always rely on a congruence criterion.
• Writing the congruence statement with vertices out of corresponding order.
• Confusing equality of shape (similarity) with equality of size (congruence).

Real-life relevance

Congruence ideas help in engineering (identical parts), construction (matching triangular supports), manufacturing (identical components), tiling patterns, and art/design (repeating identical shapes).