Important Formula: A Tale of Three Intersecting Lines | Mathematics (Ganita Prakash) Class 7 - New NCERT PDF Download

Triangle Inequality (All three must be satisfied):

For any triangle with sides a, b, c:

• a + b > c

• b + c > a

• c + a > b

Equilateral Triangle Construction:

Given side length s, to construct an equilateral triangle:

1. Draw a base of length s.

2. From each endpoint, draw an arc of radius s.

3. The point where the arcs intersect is the third vertex.

Condition for a Triangle to Exist:

• The sum of the two smaller sides must be greater than the longest side.
  Sum of the two smaller sides > Longest side

Conditions for Arc Intersection (Circle Construction):

• Internal intersection (triangle possible): Sum of radii > distance between centers

• Touching (degenerate triangle): Sum of radii = distance between centers

• No intersection: Sum of radii < distance between centers

Angle Sum Property:

• The sum of interior angles in a triangle is always 180°:
  ∠A + ∠B + ∠C = 180°

Exterior Angle Property:

• An exterior angle of a triangle is equal to the sum of the two opposite interior angles:
  For example, ∠ACD = ∠A + ∠B

Third Angle of a Triangle (if two angles are known):

• Third angle = 180° − (sum of the two given angles)

Triangle Inequality Condition

For any triangle with sides a, b, and c, all of the following must be true:

• a + b > c

• b + c > a

• c + a > b

These conditions ensure that the three lengths can form a valid triangle.