Chapter Overview
Three line segments can enclose a triangle. This chapter investigates how side lengths and angles determine whether a triangle is possible and what properties every triangle shares.
This chapter belongs to Ganita Prakash, Grade 7. Focus on explaining each step, checking whether an answer is reasonable, and comparing more than one solution method.
Learning Objectives
After studying this chapter, you should be able to:
- explain and apply triangle inequality;
- explain and apply angle-sum property;
- explain and apply exterior angles;
- explain and apply classification;
- communicate the reasoning behind a solution clearly;
- check answers using estimation, substitution, or a second method.
Key Concepts
| Concept | Meaning | |---|---| | Triangle inequality | The sum of any two side lengths of a triangle must be greater than the third side. | | Angle-sum property | The three interior angles of every triangle add to 180°. | | Exterior angles | An exterior angle equals the sum of the two remote interior angles. | | Classification | Triangles may be classified by sides as scalene, isosceles or equilateral, and by angles as acute, right or obtuse. |
Detailed Explanation
Triangle inequality
The sum of any two side lengths of a triangle must be greater than the third side.
Angle-sum property
The three interior angles of every triangle add to 180°.
Exterior angles
An exterior angle equals the sum of the two remote interior angles.
Classification
Triangles may be classified by sides as scalene, isosceles or equilateral, and by angles as acute, right or obtuse.
Do not memorise a rule without testing it on examples. Ask what each number, operation, line, or symbol represents and whether the result fits the original situation.
Important Rules and Formulae
- For sides
a,b,c, all three inequalitiesa+b>c,b+c>a, andc+a>bmust hold. - An equilateral triangle has three 60° angles.
- A triangle can have at most one right or obtuse angle.
Worked Examples
Example 1
Problem: Can lengths 4 cm, 7 cm and 12 cm form a triangle?
Solution: No. 4 + 7 = 11, which is not greater than 12.
Example 2
Problem: Two angles are 48° and 67°. Find the third.
Solution: 180° − (48° + 67°) = 65°.
Show the mathematical statement, substitution or construction step before writing the final answer. Include units wherever the quantity is measured.
Common Mistakes
- Checking only one triangle inequality when the largest side is not identified.
- Using 360° as the angle sum of a triangle.
- Assuming every isosceles triangle is equilateral.
Quick Revision
- Triangle inequality: The sum of any two side lengths of a triangle must be greater than the third side.
- Angle-sum property: The three interior angles of every triangle add to 180°.
- Exterior angles: An exterior angle equals the sum of the two remote interior angles.
- Classification: Triangles may be classified by sides as scalene, isosceles or equilateral, and by angles as acute, right or obtuse.
Practice Questions
- Can 6 cm, 8 cm and 10 cm form a triangle?
- Find the third angle if two are 90° and 35°.
- What are the angles of an equilateral triangle?
- An exterior angle is 120° and one remote interior angle is 50°. Find the other.
Answers and Explanations
- Yes; all triangle inequalities hold.
- 55°.
- 60°, 60°, 60°.
- 70°.
Self-Check
Explain one rule from this chapter in your own words, create a fresh example, solve it, and verify the answer. If the explanation and verification agree, the concept is understood rather than merely memorised.