Triangles and its Properties for Class 9

A triangle is a fundamental geometric shape that holds a special place in the realm of mathematics. Triangles and their properties find practical applications in various fields. Their properties, classifications and theorems contribute significantly to various mathematical concepts and applications. By understanding the concepts based on triangles, we unlock a world of mathematical possibilities and lay the foundation for further exploration in the field of geometry. In this chapter, we delve into the triangles and explore their inherent properties, the midpoint theorem and the Pythagoras theorem.

Triangles and their Classification

A triangle is a polygon which is characterized by three sides, three vertices and three angles. The classification of triangles based on sides and angles is as follows:

Angle Sum Property of a Triangle

The Angle Sum Property of a triangle states that the total of its angles is 180° which is equivalent to the measure of two right angles.

Exterior Angle of a Triangle

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

Congruence

Congruence is a term used to describe objects that are like mirror images of each other – their shapes and sizes match. The ducks are congruent with each other which is shown as:

Congruence of Triangles

Congruence of triangles means that two triangles are identical. They are the same size and shape. To be considered congruent, all three sides and all three angles of one triangle must be equal to the corresponding sides and angles of the other triangle.

Congruent Triangles

Two triangles are said to be congruent if every angle of one triangle is equal to the corresponding angle of the other and every side of one is equal to the corresponding side of the other. Triangles that are congruent can be moved, rotated or flipped to exactly overlap with each other.

The symbol '≅' is used to show that two triangles are congruent.

Note: If any three entities, either angles or sides, of congruent triangles are equal, then the remaining three entities are also equal by using CPCT theorem. Corresponding parts of the congruent triangles (CPCT) states that if two triangles are congruent, then their corresponding parts (angles and sides) are also congruent.

Criteria for Congruence of Triangles

Here are the five criteria that tell us when two triangles are congruent:

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

If AB = DE, BC = EF and CA = FD, then △ABC ≅ △DEF by SSS.
If the triangles are congruent, then their corresponding angles are also equal
(∠A = ∠D, ∠B = ∠E and ∠C = ∠F) by CPCT.

2. Side-Angle-Side (SAS): If two sides and the angle between them of one triangle are equal to the corresponding sides and angle of another triangle, then the triangles are congruent.

If AB = DE, ∠B = ∠E and BC = EF, then △ABC ≅ △DEF by SAS.
If the triangles are congruent, then their corresponding angles and sidesare also equal (CA = FD, ∠A = ∠D and ∠C = ∠F) by CPCT.

3. Angle-Side-Angle (ASA): If two angles and the side included between those angles of one triangle are equal to the corresponding angles and side of another triangle, then the triangles are congruent.

If ∠B = ∠E, BC = EF and ∠C = ∠F, then △ABC ≅ △DEF by ASA.
If the triangles are congruent, then their corresponding angles and sides are also equal (CA = FD, AB = DE and ∠A = ∠D) by CPCT.

4. Angle-Angle-Side (AAS): If two angles and the side that is not included between those angles of one triangle are equal to the corresponding angles and side of another triangle, then the triangles are congruent.

If ∠B = ∠E, ∠C = ∠F and CA = FD, then △ABC ≅ △DEF by AAS.

If the triangles are congruent, then their corresponding angles and sides are also equal (BC = EF, AB = DE and ∠A = ∠D) by CPCT.

5. Right Hypotenuse Side (RHS): For right-angled triangles, if the hypotenuse and one leg of one triangle are equal to the hypotenuse and the corresponding leg of another triangle, then the triangles are congruent.

If ∠B = ∠E = 90°, CA = FD and AB = EF, then △ABC ≅ △DEF by RHS.
If the triangles are congruent, then their corresponding angles and sides are also equal (BC = DE, ∠A = ∠F and ∠C = ∠D) by CPCT.

Note: The SSA (Side-Side-Angle) and AAA (Angle-Angle-Angle) conditions are not valid criteria for proving congruence in all cases in Euclidean geometry.

SSA (Side-Side-Angle): Two sides and an angle opposite to them are equal in two triangles might not guarantee congruence. If the angle is a right angle or obtuse, there may be no solution and congruence is not assured. If the angle is acute, there may be two possible triangles satisfying the SSA condition, which is shown as

Angle-Angle-Angle (AAA): The Angle-Angle-Angle (AAA) criterion states that if all three angles of one triangle are equal to the corresponding angles of another triangle, the triangles are congruent. However, this criterion is not always sufficient for guaranteeing congruence because it does not account for the size of the triangles. Two triangles sharing identical angles (AAA) may not necessarily have the same size or dimensions.
The equilateral triangles are not congruent even if all the angles are equal because the corresponding sides are not equal, which is shown as

These criteria help us determine when two triangles can be considered identical and ensure they have the same shape and size.

Important Keys to be Remembered

1. The sum of interior angles of triangle is equal to 180°.
2. If a side of a triangle is extended, the exterior angle formed is equal to the sum of the two opposite interior angles.
3. If two sides of a triangle are the same length, then the angles across from them are also the same. [Angles on opposite sides of equal length in a triangle are equal.]
4. If two angles in a triangle are the same, then the sides across from them are also the same. [Sides opposite to equal angles in a triangle are equal.]
5. If the altitudes of a triangles are equal, then it is equilateral. 
6. In a triangle where all sides are equal (equilateral), each angle is 60°.
7. If the bisector of the vertical angle of a triangle bisects the opposite side, then the triangle is isosceles.
8. A triangle has three medians, each connecting a vertex to the midpoint of the opposite side.
9. An altitude of a triangle is a line segment drawn from a vertex perpendicular to the opposite side. This altitude forms a right angle with the side it intersects.
10. The sum of any two sides of a triangle is always greater than the third side.
    
11. The difference between any two sides of a triangle is less than the third side.
    
12. If one angle is larger than another in a triangle, then the side opposite the larger angle is longer.
13. If one side is longer than another in a triangle, then the angle opposite the longer side is larger.

Theorems Based On Triangles

Theorem 1: Mid-point Theorem 

The Midpoint Theorem states that in a triangle, if the midpoints of two sides are connected with a line segment that is parallel to the third side and is equal to half the length of the third side.

If D is the midpoint of AB and E is the midpoint of AC, then the line segment EF is parallel to the third side BC and is half the length of BC.

Theorem 2: Converse of Mid-point Theorem 

The Converse of the Midpoint Theorem states that if a line segment in a triangle joins the midpoint of one side to the midpoint of another side and is parallel to the third side, then it divides the third side into two equal segments.

Theorem 3: Pythagoras' Theorem

Pythagoras' Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Note:

Theorem 3: Converse of Pythagoras' Theorem

Converse of Pythagoras' Theorem states that if in a triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right-angled triangle.