Vardaan
Class 7 Maths • Chapter 06

The Triangle and Its Properties

Vardaan Learning Institute • Detailed Notes with Practice Questions

🔺 1. Types of Triangles

A. Based on Sides

All sides equal EquilateralAll 3 sides equal
All angles = 60°
2 sides equal Isosceles2 sides equal
2 base angles equal
All different ScaleneAll 3 sides different
All 3 angles different

B. Based on Angles

All angles < 90° Acute TriangleAll three angles are acute
Right TriangleOne angle = 90°
Has hypotenuse (longest side)
One angle > 90° Obtuse TriangleOne angle is obtuse (>90°)

📐 2. Properties of Triangles

Angle Sum Property: ∠A + ∠B + ∠C = 180° (sum of angles of any triangle)
Exterior Angle Property: Exterior angle = sum of two non-adjacent interior angles
Triangle Inequality: Sum of any two sides > third side
Exterior Angle Property — Detailed:
If one side of a triangle is extended, the angle formed outside (exterior angle) = sum of the two opposite interior angles.
Example: If ∠A = 50° and ∠B = 70°, the exterior angle at C = 50° + 70° = 120° (NOT 180°−60°=120°... well same answer, but the property connects it directly to remote interior angles!)
Triangle Inequality — Can a triangle be formed?
Add the two smaller sides. If their sum > third side, a triangle CAN be formed.
• Sides 5, 8, 12: 5+8=13 > 12 ✓ → Triangle possible
• Sides 3, 4, 8: 3+4=7 < 8 ✗ → Triangle NOT possible
• Sides 6, 6, 12: 6+6=12, NOT greater → Degenerate (flat line), NOT a valid triangle ✗

📏 3. The Pythagoras Theorem (for Right Triangles)

b (base) a c (hypotenuse) b² a² c² = a² + b² Hypotenuse² = Sum of squares of other two sides
Common Pythagorean Triplets (must memorise!):
• 3, 4, 5 (3²+4²=9+16=25=5²) ✓
• 5, 12, 13 (25+144=169=13²) ✓
• 8, 15, 17 (64+225=289=17²) ✓
• 7, 24, 25 ✓
Multiples work too: (6, 8, 10), (9, 12, 15), (10, 24, 26)

🔺 4. Medians, Altitudes, and Special Lines

Special Line Definition Meeting Point Property
Median Line from a vertex to the midpoint of the opposite side Centroid (divides median in 2:1 ratio) All 3 medians bisect each other's length in 2:1
Altitude Perpendicular from a vertex to the opposite side (or its extension) Orthocentre Can be inside (acute), on vertex (right), or outside (obtuse) the triangle
Perpendicular Bisector Perpendicular to a side at its midpoint Circumcentre Equidistant from all three vertices
Angle Bisector Bisects an angle of the triangle Incentre Equidistant from all three sides
✏️ Practice Questions — Triangle Properties (28 Questions)

Section A — Basics Easy

Q1. Two angles of a triangle are 50° and 70°. Find the third.
Q2. Can a triangle have angles 90°, 95°, and 5°? Why?
Q3. Can sides 4, 7, 11 form a triangle?
Q4. Classify 6, 6, 8 by sides
Q5. Find hypotenuse if legs are 9 and 12
Q6. Is 8, 15, 17 a right-angled triangle?
Q7. Exterior angle = 110°. One interior angle = 45°. Find the other interior angle.
Q8. Angles of a triangle are in ratio 1:2:3. Find all three.

Section B — Medium Medium

Q9. In a right triangle, one acute angle is 35°. Find the other acute angle.
Q10. An equilateral triangle has perimeter = 42 cm. Find area using formula A = (√3/4)s².
Q11. In △ABC, ∠A = 2∠B = 3∠C. Find all angles.
Q12. An isosceles triangle has apex angle 40°. Find the base angles.
Q13. In isosceles △PQR with PQ = PR, ∠Q = 65°. Find ∠P.
Q14. A ladder 13 m long leans against a wall. If foot is 5 m from wall, how high does it reach?
Q15. Legs of a right triangle are in ratio 3:4. Hypotenuse is 25 cm. Find the legs.
Q16. In △ABC, exterior angle at B is 110° and ∠A = 50°. Is the triangle right-angled?

Section C — Applied / Higher Hard

  1. Q17. Two sides of a triangle are 7 cm and 10 cm. The third side must be greater than ___ and less than ___ cm.
  2. Q18. A rectangular field is 24 m × 10 m. Find the length of a diagonal path across the field.
  3. Q19. In △PQR, ∠P = x, ∠Q = 2x, ∠R = 3x. Find x. What type of triangle is it?
  4. Q20. In △ABC, ∠A is exterior angle's supplement. ∠A = 70°. Find the exterior angle at A. Then find ∠B if ∠B = ∠C.
  5. Q21. A right triangle has hypotenuse = 41 cm and one leg = 9 cm. Find the other leg. Hence check if it's a pythagorean triplet.
  6. Q22. In triangle ABC, D is midpoint of BC. AD is a median. If AD = 12 cm, how far is the centroid G from D?
  7. Q23. The angles of a triangle are in AP (arithmetic progression) with common difference 15°. Find all three angles.
  8. Q24. Is a triangle with sides √3, √4, √5 a right triangle? Verify.
  9. Q25. An altitude from vertex A of △ABC has length 8 cm, and base BC = 15 cm. What is the area of △ABC?
  10. Q26. In △XYZ, ∠X = 90°. XZ = 8, YZ = 17. Find XY. Then find sin Y, cos Y, tan Y.
  11. Q27. The exterior angle of a triangle is 4 times one of its remote interior angles. The other remote interior angle is 40°. Find all angles of the triangle.
  12. Q28. Four points A(0,0), B(3,0), C(3,4), D(0,4) form a rectangle. Find the length of diagonal AC using distance formula = Pythagoras theorem.
✅ Key Answers: Q1: 60° | Q2: No (sum=190°>180°) | Q3: No (4+7=11 not greater) | Q4: Isosceles | Q5: 15 | Q6: Yes | Q7: 65° | Q8: 30°,60°,90° | Q14: 12m | Q15: 15cm, 20cm | Q17: greater than 3, less than 17 | Q19: 30° (right triangle at R=90°) | Q25: 60 cm²

📝 Quick Revision

  1. Sum of angles of triangle = 180° (Angle Sum Property)
  2. Exterior angle = Sum of two non-adjacent interior angles
  3. Triangle Inequality: Sum of any two sides > third side (to check if a triangle can be formed)
  4. Pythagoras Theorem: c² = a² + b² (only for right-angled triangle)
  5. Common triplets: 3-4-5, 5-12-13, 8-15-17, 7-24-25
  6. Medians meet at Centroid (2:1 ratio); Altitudes at Orthocentre
  7. Perpendicular bisectors at Circumcentre; Angle bisectors at Incentre