Triangles Class 9 ||Maths|| Chapter 7 NCERT Notes

Triangles ||Maths|| Chapter 7 NCERT Notes

1. Basics of Triangles

A triangle has three sides, three angles, and three vertices. The sum of the interior angles of a triangle is always 180°.

Types of Triangles Based on Sides:

  • Equilateral Triangle: All sides are equal, and each angle is 60°.
  • Isosceles Triangle: Two sides are equal, and the angles opposite these sides are equal.
  • Scalene Triangle: All sides are of different lengths, and all angles are of different measures.

Types of Triangles Based on Angles:

  • Acute Triangle: All angles are less than 90°.
  • Right Triangle: One angle is exactly 90°.
  • Obtuse Triangle: One angle is more than 90°.

2. Congruence of Triangles

Two triangles are said to be congruent if all the corresponding sides and angles are equal. When two triangles are congruent, they have the same shape and size. Congruence is represented by the symbol ≅.

Criteria for Triangle Congruence:

  1. SAS (Side-Angle-Side) Congruence Rule: If two sides and the angle included between them in one triangle are equal to the corresponding two sides and the included angle in another triangle, the triangles are congruent.

  2. ASA (Angle-Side-Angle) Congruence Rule: If two angles and the side included between them in one triangle are equal to the corresponding two angles and the included side in another triangle, the triangles are congruent.

  3. AAS (Angle-Angle-Side) Congruence Rule: If two angles and a non-included side of one triangle are equal to the corresponding angles and non-included side of another triangle, the triangles are congruent.

  4. SSS (Side-Side-Side) Congruence Rule: If all three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent.

  5. RHS (Right Angle-Hypotenuse-Side) Congruence Rule: In right-angled triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of another triangle, then the triangles are congruent.

3. Important Theorems Related to Triangles

Theorem 1: Angle Sum Property of a Triangle

The sum of the interior angles of a triangle is 180°.

Theorem 2: Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Theorem 3: Isosceles Triangle Theorem

In an isosceles triangle, the angles opposite the equal sides are equal.

Theorem 4: Converse of Isosceles Triangle Theorem

If two angles of a triangle are equal, then the sides opposite to these angles are also equal.

4. Inequalities in a Triangle

  1. Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

    AB+BC>CA, AB + CA>BC, BC + CA>AB

  2. Exterior Angle Inequality Theorem: The measure of an exterior angle of a triangle is greater than either of the opposite interior angles.

5. Right-Angled Triangle and Pythagoras Theorem

In a right-angled triangle, the side opposite the right angle is called the hypotenuse. The other two sides are called the legs.

Pythagoras Theorem:

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If c is the hypotenuse and a and b are the other two sides, then:

c2=a2+b2

Converse of Pythagoras Theorem:

If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.

6. Properties of Medians, Altitudes, and Angle Bisectors

  • Median: A line segment joining a vertex to the midpoint of the opposite side. The three medians of a triangle meet at a point called the centroid, which divides each median in the ratio 2:1.

  • Altitude: A perpendicular segment from a vertex to the opposite side (or the line containing the opposite side). The point where the three altitudes meet is called the orthocenter.

  • Angle Bisector: A line that divides an angle of a triangle into two equal parts. The three angle bisectors meet at a point called the incenter, which is equidistant from all the sides of the triangle.

Summary:

  • A triangle is defined by its sides and angles, and it has various properties that determine its shape and size.
  • Congruent triangles have equal corresponding sides and angles and can be proven by different criteria like SSS, SAS, ASA, etc.
  • Important theorems such as the angle sum property, isosceles triangle theorem, and Pythagoras theorem are foundational for solving problems related to triangles.
  • Properties related to medians, altitudes, and angle bisectors help in understanding the internal structure of triangles.