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Give a Class 7 student three sticks of 10 cm, 15 cm and 30 cm and ask for a triangle. They will try, and fail, and that failure is the start of NCERT's new triangles chapter. This post covers the triangle inequality, the angle sum property and exterior angles, altitudes and the types of triangles, with worked examples, common mistakes and a practice routine.
What is in the Class 7 triangles chapter?
In NCERT's Ganita Prakash for Class 7, Part 1, Chapter 7 is "A Tale of Three Intersecting Lines". The Part 1 contents and introduction say it explores triangles and results related to the lengths of their sides and their angles. The full chapter is a free PDF: A Tale of Three Intersecting Lines. It builds equilateral triangles with a compass, constructs triangles from sides and angles, proves the angle sum property, constructs altitudes, and names types of triangles. It relies on the parallel-line facts from Chapter 5, so revise those first if they feel shaky.
When do three lengths make a triangle?
Walking straight from one corner to another is always shorter than going round through the third corner. That is the triangle inequality. The chapter states it this way: three lengths satisfy the triangle inequality when each length is smaller than the sum of the other two, and then a triangle with those sides exists.
The quick test is to check only the longest side against the sum of the other two.
Worked example 1. Which sets make a triangle?
3, 4, 5: longest is 5, and 3 + 4 = 7. Since 7 > 5, yes.
10, 15, 30: longest is 30, and 10 + 15 = 25. Since 25 < 30, no.
4, 6, 10: 4 + 6 = 10, which equals the longest side. The two shorter sides lie flat along the longest, so no triangle forms.
5, 8, 12: 5 + 8 = 13 > 12, so yes.
Worked example 2. Two sides are 5 cm and 9 cm. What can the third side be? It must be less than 5 + 9 = 14 cm. It must also be more than 9 − 5 = 4 cm, or else 5 plus the third side would not beat 9. So the third side is between 4 cm and 14 cm, not including either.
Why do the angles of a triangle add up to 180°?
The chapter proves it with a parallel line. Take triangle ABC and draw a line through A parallel to BC. At A, three angles now sit along a straight line, so they add to 180°. The left one equals angle B and the right one equals angle C, because they are alternate angles across parallel lines. The middle one is angle A. So A + B + C = 180°, for every triangle.
Worked example 3 (from the chapter). In triangle ABC, angle A = 50° and angle B = 60°. Side BC is extended to D. Find angle ACB and the exterior angle ACD.
50° + 60° + angle ACB = 180°, so angle ACB = 70°.
Angle ACB and angle ACD form a straight angle, so angle ACD = 180° − 70° = 110°.
Notice that 110° = 50° + 60°. The chapter asks you to explore this. The exterior angle always equals the sum of the two opposite interior angles, because both make up what is left of 180°.
Worked example 4 (with letter-numbers). The angles of a triangle are x, 2x and 3x. Then x + 2x + 3x = 6x = 180°, so x = 30°. The angles are 30°, 60° and 90°, a right-angled triangle.
The chapter also asks when two angles can belong to a triangle. Its answer: when their sum is less than 180°. Two angles of 100° and 90° add to 190°, so no triangle has both.
What is an altitude?
An altitude is a perpendicular line segment from a vertex to the opposite side. When we say the height of a triangle, we usually mean the altitude to whichever side we take as the base. In some triangles the foot of the altitude lands outside the triangle. The chapter handles this by extending the base and dropping the perpendicular onto the extended line. You can also find an altitude by paper folding: fold the base onto itself so the crease passes through the top vertex.
What are the types of triangles?
By sides: equilateral (all three equal), isosceles (two equal) and scalene (all different). By angles: acute-angled (all angles less than 90°), right-angled (one angle of 90°) and obtuse-angled (one angle more than 90°). The angle sum explains why a triangle can never have two right angles: 90° + 90° already uses up 180°.
Which mistakes come up most?
Checking only one pair of sides. Always test the longest side against the other two.
Forgetting the "equal" case. If the two shorter sides add to exactly the longest, there is no triangle.
Adding the exterior angle to all three angles. The exterior angle equals the two opposite interior angles, not the one next to it.
Drawing altitudes to the midpoint. An altitude must meet the base at a right angle, which is usually not at its midpoint.
A practice routine
Five sets of three lengths. Decide which make triangles, then build two with a ruler and compass to check.
Two angle-sum problems and one exterior-angle problem, with the reason written beside each step.
One altitude by paper folding and one by construction for an obtuse-angled triangle.
Mix in older chapters. Add a parallel-lines question and a decimals question to each set. Rohrer and Taylor (2007) found that mixing problem types gave much better scores a week later than doing one type at a time.
When you can do the steps but are unsure which fact to use first, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. For the angle facts this chapter depends on, see our guide to parallel and intersecting lines in Class 7. Congruent triangles come next, in Class 9.
This week, cut three straws to 4 cm, 6 cm and 10 cm and see for yourself why they will not close into a triangle.
