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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.

1.3 Angles of Isosceles and Equilateral Triangles

This section uses congruence to explore properties of isosceles and equilateral triangles. An isosceles triangle has two equal sides. For ΔABC with AB = AC and ∠A = 80°, the altitude AD is constructed perpendicular to BC. Triangles ADB and ADC are right-angled and share side AD. Since AB = AC and AD is common, by RHS condition, ΔADB ≅ ΔADC. Hence, corresponding angles ∠B and ∠C are equal. This proves that in an isosceles triangle, angles opposite equal sides are equal. Using this fact, the measures of ∠B and ∠C can be found by subtracting ∠A from 180° and dividing the remainder equally. For equilateral triangles, all sides are equal, so by the above property, all angles opposite these sides are equal. Since the sum of angles in a triangle is 180°, each angle in an equilateral triangle measures 60°. This is verified by construction. The section concludes by showing examples of congruent triangles in real life, such as the Louvre Museum's glass pyramids, Egyptian pyramids, dome designs, rangoli patterns, and the Howrah Bridge. These examples illustrate the practical and aesthetic importance of congruent triangles in architecture and design.

📊 Diagram: Diagrams show isosceles triangle ABC with altitude AD, right triangles ADB and ADC; equilateral triangle with equal sides and angles marked; photographs of architectural structures illustrating congruent triangles.

🧪 Activity: Identify congruent triangles in given real-life structures and patterns; verify angle equality in isosceles and equilateral triangles by construction.

🔗 Connection: Leads to exercises and problems applying congruence conditions and properties of triangles.

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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