**Notes of chapter: Triangles are presented below. Indepth notes along with worksheets and NCERT Solutions of Class 9.**

**(1) Triangle-**

A **triangle** is a simple closed curve made of three line segments. It has three vertices, three sides and three angles.

**(2) Median of a line-**

A** median is a line which** connects a vertex of a triangle to the mid- point of the opposite side.

**(3) Altitude-**

The perpendicular line segment from a vertex of a triangle to its opposite side is called an **altitude** of the triangle. A triangle has 3 altitudes.

**(4) Exterior angle-**

The **exterior angle** is the angle between any side of a shape, and a line extended from the next side.

**(5) Exterior angle property of a triangle-**

An exterior angle of a triangle is equal to the sum of its interior opposite angles. This relation is referred to as the **exterior angle property of a triangle**.

**(6) Angle sum property of a triangle-**

The total measure of the three angles of a triangle is 180^{o}. It is known as **angle sum property of a triangle.**

**(7) Equilateral triangle-**

A triangle in which all the three sides are of equal lengths is called an **equilateral triangle.**

**(8) Isosceles triangle-**

A triangle in which two sides are of equal lengths is called an **isosceles** **triangle**.

**(9)Right-angled triangle-**

A triangle which has a right angle is called** right-angled triangle.** The side opposite to the right angle (90^{0}) is called the **hypotenuse **and the other sides are known as the **legs** of the right- angled triangle.

**(10)** The sum of the lengths of any two sides of a triangle is greater than the third side.

**(11)** The difference of the lengths of any two sides of a triangle is lesser than the third side.

**(12)** **Pythagoras property-**

In a right- angled triangle,

The square on the hypotenuse = sum of the squares on the legs.

**(14) Congruence-**

**(i)**Two objects are **congruent** if they have the same dimensions and shape. In simple words, they have mirror images.

**Eg:-**

Both stars are congruent because they have mirror image.

**(ii)** Two **line segments are congruent** if they have the same length.

**Eg:-**

**(iii)** **Two angles are congruent** if they have the same (ie, equal) measure.

**Eg:-**

**(iv)Two triangles are congruent**, when their corresponding parts (ie, angles and sides) that match one another are equal.

**Eg:-**

**(15)** **Criteria for congruence of triangle-**

**(i)** **SSS congruence criterion**:

When the three sides of one triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent. This property is called as **SSS congruence of triangle.**

**Eg:-**

**(ii)SAS Congruence Criterion-**

When two sides and the angle included between them of a triangle are equal to two corresponding sides and the angle included between them of another triangle, then the triangles are congruent. This property is called as **SAS congruence of triangle.**

**Eg:-**

**(iii)** **ASA Congruence criterion-**

When two angles and the included side of a triangle are equal to two corresponding angles and the included side of another triangle, then the triangles are congruent. This property is called as **ASA congruence of triangle.**

**Eg:-**

**(iv)** **RHS congruence criterion-**

When the hypotenuse and one side of a right- angled triangle are respectively equal to the hypotenuse and one side of another right- angled triangle, then the triangles are congruent. This property is called as **RHS congruence of triangle.**

**Eg:-**

**(v) AAS Congruence criterion-**

When two pair of angles and one pair of side of a triangle are equal to two corresponding angles and corresponding side of another triangle, then the triangles are congruent. This property is called as **AAS congruence of triangle.**

**Eg:-**

**Helping Topics**