NCERT-aligned: The current Grade 7 Ganita Prakash Part I describes Chapter 7 as exploring striking properties of triangles through construction, especially relationships involving side lengths and angles. The book emphasises exploration, construction, discussion and mathematical reasoning. citeturn1search0turn1view0

🌟 Chapter at a Glance

1. 🔺 What is a Triangle?

A triangle is a closed figure formed by three line segments. It has 3 sides, 3 vertices and 3 interior angles. The three sides and three angles are connected: changing one part can restrict the others.

Let the vertices be A, B and C. The opposite side to angle A is BC; opposite angle B is AC; opposite angle C is AB.

2. 📐 Classifying Triangles by Sides

An equilateral triangle is also equiangular: each angle is 60°. An isosceles triangle has equal angles opposite its equal sides.

3. 📏 Classifying Triangles by Angles

A triangle cannot have two right angles or two obtuse angles because its three interior angles together make 180°.

4. 🧠 The Angle-Sum Property

For every triangle, ∠A + ∠B + ∠C = 180°.

Example: if two angles are 48° and 67°, the third angle is 180° − (48° + 67°) = 65°.

5. 🔍 Side–Angle Relationship

In a triangle, the larger side lies opposite the larger angle, and the smaller side lies opposite the smaller angle. Equal sides are opposite equal angles.

This is more than a rule to memorise: it lets us compare parts of a triangle even when exact measurements are unavailable.

6. 🚧 Can Any Three Lengths Make a Triangle?

No. The sum of the lengths of any two sides of a triangle must be greater than the third side.

This is the triangle inequality. It is extremely useful when deciding whether a proposed triangle is possible.

7. 🛠️ Constructing Triangles

Construction means creating a geometric figure accurately from given information, rather than drawing something that merely looks correct. A compass and ruler can transfer lengths and locate points precisely.

Construction idea: SSS

1. Draw one side of the required length.
2. With the compass, draw an arc from one endpoint using the second side length.
3. From the other endpoint, draw an arc using the third side length.
4. The intersection of the arcs gives the third vertex.
5. Join the vertex to the two endpoints.

The construction works only when the three lengths satisfy the triangle inequality.

8. 🧩 Worked Examples

1. Angles 55° and 75°: third angle = 50°.
2. Isosceles triangle: if the equal sides are AB and AC, then the opposite angles ∠C and ∠B are equal.
3. Possible sides 7, 10, 16: 7+10=17>16, 10+16>7 and 16+7>10, so a triangle is possible.
4. Largest angle: in a triangle with sides 5 cm, 8 cm and 11 cm, the angle opposite 11 cm is the largest.

🎯 Math Talk

1. Why can’t three sticks of lengths 2 cm, 3 cm and 5 cm form a triangle?
2. If two angles of a triangle are equal, what can you conclude about the opposite sides?
3. Can a triangle be both isosceles and right-angled? Explain.
4. Why does an SSS construction fail when one side is too long?

🧩 Interactive MCQs

1. A triangle has angles 50°, 60° and 70°. It is: A. acute B. right C. obtuse D. impossible
2. Two sides are 6 cm and 8 cm. Which could be the third side? A. 2 cm B. 3 cm C. 14 cm D. 15 cm
3. If two sides of a triangle are equal, the opposite angles are: A. equal B. supplementary C. right angles D. unequal
4. The largest angle is opposite the: A. shortest side B. longest side C. equal side only D. median
5. The sum of the angles of a triangle is: A. 90° B. 180° C. 270° D. 360°

Answers: 1-A, 2-B, 3-A, 4-B, 5-B

📌 Case-Based Challenge — The Bridge Frame

An engineer wants a triangular support using three metal rods of lengths 5 m, 7 m and 10 m.

1. Can the three rods form a triangle?
2. Which angle will be opposite the 10 m rod, relative to the other two angles?
3. If two angles are 42° and 68°, find the third.
4. Explain why triangular frames are useful for rigid structures.

🔥 HOTS / Competition Corner

1. Find all possible integer values of x for side lengths 5, 8 and x to form a triangle.
2. A triangle has angles x°, 2x° and 3x°. Find all three angles and classify the triangle.
3. Can a triangle have sides 4 cm, 4 cm and 8 cm? Give a mathematical reason, not a drawing.
4. Construct a triangle with sides 6 cm, 7 cm and 8 cm and explain why your construction is unique up to reflection.

📝 Chapter Worksheet

🏆 Mastery Checklist

• ☐ I can classify triangles by sides and angles.
• ☐ I can use the angle-sum property.
• ☐ I can relate side lengths to opposite angles.
• ☐ I can test the triangle inequality.
• ☐ I can construct a triangle from three side lengths when possible.
• ☐ I can explain my reasoning rather than only give an answer.

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