CBSE Class 9 Maths Notes: Triangles

Introduction to Triangles

Triangles are fundamental geometric shapes, forming the basis for understanding more complex figures. This chapter explores the properties of triangles, focusing on congruence and related theorems. We’ll delve into how to prove if two triangles are identical, and how this knowledge helps us solve geometry problems.

Congruence Rules

Congruent triangles have the same shape and size. Several rules help us determine if two triangles are congruent without having to compare all their sides and angles.

SAS (Side-Angle-Side) Congruence Rule

If two sides and the included angle (the angle between those two sides) of one triangle are equal to the corresponding two sides and the included angle of another triangle, then the two triangles are congruent.

Example: If $AB = DE$, $AC = DF$, and $\angle BAC = \angle EDF$, then $\triangle ABC \cong \triangle DEF$.

ASA (Angle-Side-Angle) Congruence Rule

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

Example: If $\angle ABC = \angle DEF$, $BC = EF$, and $\angle BCA = \angle EFD$, then $\triangle ABC \cong \triangle DEF$.

SSS (Side-Side-Side) Congruence Rule

If all three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent.

Example: If $AB = DE$, $BC = EF$, and $CA = FD$, then $\triangle ABC \cong \triangle DEF$.

RHS (Right angle-Hypotenuse-Side) Congruence Rule

If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, then the two triangles are congruent.

Example: In right triangles, if $AC = DF$ (hypotenuses) and $AB = DE$, then $\triangle ABC \cong \triangle DEF$.

Angles Opposite Equal Sides (Proof)

Theorem: Angles opposite to equal sides of a triangle are equal.

Proof:

1. Consider a triangle $ABC$ where $AB = AC$.
2. Draw the angle bisector $AD$ of $\angle BAC$, where $D$ lies on $BC$.
3. In $\triangle ABD$ and $\triangle ACD$:
   • $AB = AC$ (Given)
   • $\angle BAD = \angle CAD$ (By construction)
   • $AD = AD$ (Common side)
4. Therefore, $\triangle ABD \cong \triangle ACD$ (SAS congruence rule).
5. Hence, $\angle ABC = \angle ACB$ (Corresponding parts of congruent triangles – CPCT).

Sides Opposite Equal Angles (Statement)

Theorem: Sides opposite to equal angles of a triangle are equal.

This is the converse of the theorem discussed in the “Angles Opposite Equal Sides” section. If $\angle B = \angle C$ in $\triangle ABC$, then $AB = AC$. (Proof can be done using the ASA congruence rule by constructing the angle bisector or by using contradiction.)

Problem Solving Using Congruence

Congruence rules are crucial for solving various geometry problems. By identifying congruent triangles, we can deduce relationships between sides and angles, enabling us to prove theorems and solve numerical problems.

Steps for Problem Solving:

1. Identify the given information and what needs to be proved.
2. Draw a clear diagram.
3. Identify pairs of triangles that might be congruent.
4. Apply congruence rules (SAS, ASA, SSS, RHS) to prove congruence.
5. Use CPCT (Corresponding Parts of Congruent Triangles) to deduce required results.

Example: Proving the properties of an isosceles triangle, or proving equality of line segments in complex geometric figures.