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MathematicsClass 7Geometric

Geometric | Class 7 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 5 min read

Geometric – this guide gives you a concise, exam-ready overview of Geometric from Class 7 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

1.2 Congruence of Triangles

This section explores how to determine if two triangles are congruent and how to replicate a triangular frame exactly. Meera and Rabia want to make a cardboard cutout identical to a large triangular frame but cannot trace it directly. They measure the sides: 40 cm, 60 cm, and 80 cm. Meera suggests that side lengths alone are sufficient to construct a congruent triangle, without measuring angles. This is justified by the SSS (Side Side Side) condition: if two triangles have the same three side lengths, they are congruent. An example with smaller lengths (4 cm, 6 cm, 8 cm) is given. Rabia attempts construction by drawing one side and circles with radii equal to the other two sides from each endpoint, resulting in two possible triangles (ABE and ABF). These two triangles are congruent, as can be verified by tracing or symmetry arguments. The section explains conventions to express congruence, emphasizing the importance of matching corresponding vertices so that equal sides overlap when superimposed. The notation ΔABC ≅ ΔXYZ means vertex A corresponds to X, B to Y, and C to Z. Incorrect vertex order can lead to wrong conclusions. Examples with rectangles and other triangles illustrate correct vertex correspondence. The section also discusses measuring angles alone is insufficient for congruence, as triangles with equal angles can differ in size (similar triangles). The SAS (Side Angle Side) condition is introduced: two sides and the included angle equal in two triangles guarantee congruence. The SSA (Side Side Angle) condition does not guarantee congruence, as two different triangles can be constructed with the same two sides and a non-included angle. The ASA (Angle Side Angle) and AAS (Angle Angle Side) conditions are also explained as sufficient for congruence. The RHS (Right angle, Hypotenuse, Side) condition is introduced for right-angled triangles: if the hypotenuse and one side are equal, the triangles are congruent. The section concludes with a summary of all sufficient conditions for triangle congruence.

📊 Diagram: Diagrams include the large triangular frame, construction of triangles using circles from endpoints, pairs of congruent triangles with vertex correspondence, examples showing multiple triangles with same angles but different sizes, construction steps for SAS and SSA cases, right-angled triangle construction with perpendicular lines and arcs.

🧪 Activity: Construct triangles with given measurements to verify congruence or non-congruence under different conditions (SSS, SAS, SSA, ASA, RHS).

🔗 Connection: Prepares for understanding properties of special triangles and applying congruence to deduce angle properties.

Frequently asked questions

ΔABC is isosceles with AB = AC, and ∠A = 80°. What can we say about ∠B and ∠C? Construct the altitude from A to BC. Can you use this fact to find ∠B and ∠C?

Since ΔABC is isosceles with AB = AC, the angles opposite these sides are equal. Constructing the altitude AD from A to BC creates two right triangles, ΔADB and ΔADC. By RHS congruence (Right angle, Hypotenuse, Side), ΔADB ≅ ΔADC. Therefore, ∠B = ∠C. Since the sum of angles in ΔABC is 180°, we have ∠A + ∠B + ∠C = 180°. Given ∠A = 80°, and ∠B = ∠C, let each be x. Then 80° + x + x = 180°, so 2x = 100°, x = 50°. Hence, ∠B = ∠C = 50°.

What can we say about the angles of an equilateral triangle? Verify this by construction. What could be their measures?

In an equilateral triangle, all sides are equal. Using the fact that angles opposite equal sides are equal, all three angles are equal. Since the sum of angles in a triangle is 180°, each angle measures 180° ÷ 3 = 60°. This can be verified by constructing an equilateral triangle and measuring its angles.

1. ΔAIR ≅ ΔFLY. Identify the corresponding vertices, sides and angles. 2. Each of the following cases contains certain measurements taken from two triangles. Identify the pairs in which the triangles are congruent to each other, with reason. Express the congruence whenever they are congruent. (a) AB = DE, BC = EF, CA = DF (b) AB = EF, ∠A = ∠E, AC = ED (c) AB = DF, ∠B = ∠D = 90°, AC = FE (d) ∠A = ∠D, ∠B = ∠E, AC = DF (e) AB = DF, ∠B = ∠F, AC = DE 3. It is given that OB = OC, and OA = OD. Show that AB is parallel to CD. [Hint: AD is a transversal for these two lines. Are there any equal alternate angles?] 4. ABCD is a square. Show that ΔABC ≅ ΔADC. Is ΔABC also congruent to ΔCDA? Give more examples of two triangles where one triangle is congruent to the other in two different ways, as in the case above. Can you give an example of two triangles where one is congruent to the other in six different ways? 5. Find ∠B and ∠C, if A is the centre of the circle. 6. Find the missing angles. As per the convention that we have been following, all line segments marked with a single ‘|’ are equal to each other and those marked with a double ‘|’ are equal to each other, etc.

1. Corresponding vertices: A ↔ F, I ↔ L, R ↔ Y. Corresponding sides: AI ↔ FL, IR ↔ LY, AR ↔ FY. Corresponding angles: ∠A ↔ ∠F, ∠I ↔ ∠L, ∠R ↔ ∠Y.

2. (a) AB=DE, BC=EF, CA=DF → SSS congruence → ΔABC ≅ ΔDEF. (b) AB=EF, ∠A=∠E, AC=ED → SAS congruence → ΔABC ≅ ΔEFD. (c) AB=DF, ∠B=∠D=90°, AC=FE → RHS congruence → ΔABC ≅ ΔDFE. (d) ∠A=∠D, ∠B=∠E, AC=DF → ASA congruence → ΔABC ≅ ΔDEF. (e) AB=DF, ∠B=∠F, AC=DE → Not sufficient for congruence (two angles and non-included side).

3. Given OB=OC and OA=OD, cons

Draw lines and split the region consisting of white squares into 6 smaller congruent regions.

To split the given region of white squares into 6 smaller congruent regions, draw lines such that each smaller region is congruent in shape and size. This can be done by dividing the figure symmetrically, ensuring equal areas and congruent shapes. The exact lines depend on the figure shown, but typically involve drawing lines parallel or at angles that create equal partitions.

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