Chapter focus
A triangle’s three sides and three angles are connected. Angle sums, equal-side relationships and the triangle inequality help decide what shapes are possible and calculate missing measures.
These original notes support the investigations named in NCERT CIET resources for Ganita Prakash Part I; use the textbook’s diagrams and activities for the complete treatment.
1. Interior angles of a triangle
The three interior angles of every triangle add to 180°. This can be seen by drawing a line through one vertex parallel to the opposite side: alternate interior angle relationships show that the three angles make a straight angle.
Worked example
A triangle has angles 48° and 67°. Its third angle is 180° − 48° − 67° = 65°.
An exterior angle equals the sum of the two opposite interior angles. If those angles are 48° and 67°, the exterior angle is 115°.
2. Isosceles triangles
An isosceles triangle has at least two equal sides. The angles opposite those equal sides are equal. Conversely, if two angles of a triangle are equal, the sides opposite them are equal.
Worked example
An isosceles triangle has a vertex angle of 40°. The two equal base angles share 180° − 40° = 140°, so each is 70°.
When identifying the equal angles, match each angle to the side opposite it; equal sides do not meet at the equal base angles.
3. Decide whether three lengths can form a triangle
The sum of any two side lengths must be greater than the third side. It is enough to check that the two shorter lengths add to more than the longest.
Example: 4 cm, 6 cm and 9 cm can form a triangle because 4 + 6 = 10 > 9. But 3 cm, 5 cm and 8 cm cannot: 3 + 5 = 8, so the sides would lie flat instead of enclosing a triangle.
4. Use a labelled sketch, not appearance
Mark equal sides, known angles and the quantity to find. A sketch helps organize information, but it may not be drawn to scale. Use a stated property—angle sum, equal sides or triangle inequality—to justify the conclusion.
Common errors to avoid
- A triangle’s interior angles total 180°, not 360°.
- Equal sides are opposite equal angles in an isosceles triangle.
- The sum of two shorter sides must be strictly greater than the longest side.
- Do not conclude two sides are equal just because they look equal in a sketch.
Practice: calculate or decide
1. Find the third angle of a triangle with angles 55° and 72°.
Answer: 180° − 55° − 72° = 53°.
2. An isosceles triangle has a vertex angle of 36°. Find each base angle.
Answer: (180° − 36°) ÷ 2 = 72°.
3. Can lengths 5 cm, 7 cm and 13 cm form a triangle?
Answer: No. The two shorter sides sum to 12 cm, which is less than 13 cm.
4. Two angles of a triangle are 40° and 40°. What can you conclude about the opposite sides?
Answer: They are equal; the triangle is isosceles.
5. A triangle has angles 35°, 65° and 80°. Find an exterior angle at the 80° vertex.
Answer: 180° − 80° = 100°, or 35° + 65° = 100°.
Chapter summary
Triangle angles total 180°. Equal sides face equal angles, and equal angles face equal sides. Three lengths form a triangle only when each pair sums to more than the remaining side. Use these facts with a labelled diagram to justify each step.
Open Ganita Prakash Part I on NCERT · NCERT CIET Grade 7 resources