Triangles| Properties of Triangle| Congruence of Triangle

Notes of chapter: Triangles| Properties of Triangle| Congruence of Triangle 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.

 

Triangle of NCERT chapter Triangle

(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.

Median of NCERT chapter Triangles

(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.

Altitude of chapter Triangles

(4) Exterior angle-

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

Exterior angle of NCERT chapter Triangles

(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.

Exterior angle property of a triangle of NCERT chapter Triangles

(6) Angle sum property of a triangle-

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

 

Angle sum property of a triangle of NCERT chapter Triangles

(7) Equilateral triangle-

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

Equilateral triangle of NCERT chapter Triangles

(8) Isosceles triangle-

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

Isosceles triangle of NCERT chapter Triangles

(9)Right-angled triangle-

A triangle which has a right angle is called right-angled triangle. The side opposite to the right angle (900) is called the hypotenuse and the other sides are known as the legs of the right- angled triangle.

Right angled triangle of NCERT chapter Triangles

(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.

 

Pythagoras theorem of NCERT chapter Triangles

Properties of a triangle of NCERT chapter Triangles

(14) Congruence-

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

Eg:-

 

 

Congruence of shape of stars of NCERT chapter Triangles

Both stars are congruent because they have mirror image.

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

Eg:-

 

 

Congruence of line segments of NCERT chapter Triangles

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

Eg:-

 

Congruence of angles of NCERT chapter Triangles

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

Eg:-

 

Congruence of triangles of NCERT chapter Triangles

 

 

(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:-

SSS Congruence criterion of NCERT chapter Triangles

 

(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:-

 

SAS congruence criterion of NCERT chapter Triangles

 

(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:-

 

ASA congruence criterion of NCERT chapter Triangles

(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:-

 

RHS congruence criterion of NCERT chapter Triangles

(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:-

 

AAS congruence criterion of NCERT chapter Triangles

Helping Topics

NCERT Solutions Class 9

Worksheet Class 9

NCERT Solutions Class 7

Worksheet Class 7

 

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