A Tale of Three Intersecting Lines
Three lines meeting two at a time enclose a triangle — the simplest closed figure. This Class 7 Ganita Prakash chapter studies triangles: their types by sides and angles, the angle sum property, the exterior angle relationship, and the triangle inequality that decides which side lengths can form a triangle.
Learning objectives
- Classify triangles by sides and angles.
- Use the angle sum property.
- Apply the exterior angle property.
- Use the triangle inequality.
Key concepts
Types of triangles
By sides, a triangle is equilateral (all sides equal), isosceles (two sides equal) or scalene (all sides different). By angles, it is acute (all angles less than 90°), right (one angle 90°) or obtuse (one angle more than 90°). Every triangle fits one description from each group.
Angle sum property
The three interior angles of any triangle always add up to 180°. So if two angles are 50° and 60°, the third is 180° − 50° − 60° = 70°. This single fact solves a huge number of triangle problems.
Exterior angle property
If one side of a triangle is extended, the exterior angle formed equals the sum of the two interior angles opposite to it. So an exterior angle of 110° equals the two far interior angles added together, which is often quicker than using the angle sum.
Triangle inequality
Not any three lengths form a triangle. The triangle inequality says the sum of the lengths of any two sides must be greater than the third side. So sides 3, 4 and 5 work (3 + 4 > 5), but 2, 3 and 6 do not (2 + 3 < 6).
Important formulas
Angle sum
angle1 + angle2 + angle3 = 180°
Exterior angle
exterior angle = sum of two opposite interior angles
Triangle inequality
sum of any two sides > third side
Key definitions
- Equilateral triangle
- A triangle with all three sides (and angles) equal.
- Isosceles triangle
- A triangle with two equal sides.
- Exterior angle
- The angle formed outside a triangle when a side is extended.
- Triangle inequality
- The rule that any two sides together exceed the third.
Solved examples
Q1. Two angles of a triangle are 40° and 75°. Find the third.
Solution: 180° − 40° − 75° = 65°.
Q2. Can sides 2 cm, 3 cm and 6 cm form a triangle?
Solution: No, because 2 + 3 = 5 is not greater than 6.
Q3. An exterior angle is 120°; the two opposite interior angles are 70° and ?.
Solution: 120° − 70° = 50°.
Common mistakes to avoid
- Thinking the angles can add to something other than 180°.
- Confusing isosceles (two equal sides) with equilateral (all equal).
- Forgetting to check the triangle inequality before forming a triangle.
- Adding the wrong interior angles for the exterior angle.
A Tale of Three Intersecting Lines — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
The angles of a triangle add up to:
Practice questions
Short answer
What is the angle sum of a triangle?
180°.
Name the three types of triangle by sides.
Equilateral, isosceles and scalene.
State the triangle inequality.
The sum of any two sides is greater than the third side.
Long answer
State and use the angle sum and exterior angle properties of a triangle.
The angle sum property says the three interior angles of any triangle add up to 180°, so if two angles are known the third is found by subtracting their sum from 180°. The exterior angle property says that when a side is extended, the exterior angle equals the sum of the two interior angles opposite to it. For example, in a triangle with interior angles 50° and 60°, the third angle is 180° − 50° − 60° = 70°, and the exterior angle at that vertex equals 50° + 60° = 110°. The two properties are linked, since the exterior angle and its adjacent interior angle form a linear pair (110° + 70° = 180°).
Explain the triangle inequality and why some sets of lengths cannot form a triangle.
The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is because the two shorter sides must be long enough to 'reach across' and meet over the longest side. For example, sides 3, 4 and 5 form a triangle because 3 + 4 = 7 is greater than 5. But sides 2, 3 and 6 cannot, because 2 + 3 = 5 is less than 6 — the two shorter sides are too short to meet, so the figure cannot close into a triangle.
HOTS (Higher Order Thinking)
Can a triangle have two right angles? Explain.
No. Two right angles already total 180°, leaving 0° for the third angle, which is impossible.
An isosceles triangle has a vertex angle of 40°. What are its base angles?
The base angles are equal and together make 180° − 40° = 140°, so each is 70°.
Quick revision
Revision notes
- By sides: equilateral, isosceles, scalene; by angles: acute, right, obtuse.
- Angle sum = 180°.
- Exterior angle = sum of two opposite interior angles.
- Triangle inequality: any two sides together > third side.
Key takeaways
- A triangle's angles always total 180°.
- The exterior angle equals the two far interior angles.
- Side lengths must satisfy the triangle inequality.
Frequently asked questions
Do all triangles have an angle sum of 180°?
Yes, the interior angles of every triangle add to 180°.
What is an isosceles triangle?
A triangle with two equal sides (and two equal base angles).
When can three lengths form a triangle?
When the sum of any two of them is greater than the third.