Triangles is a closed figure with three straight sides and three angles. Those triangles which have exactly the same three sides and exactly the same three angles. If the figures have exactly the same shape and size are called congruent figures. The figures which are exactly the same in the shape but sizes may be different are known as similar figures.
Triangles is a closed figure with three straight sides and three angles.
Properties of Triangle
The sum of the length of any two sides of a triangle is always greater than the length of its third side. On \(\triangle\)PQR PQ + QR > PR, QR + PR > PQ and PQ + PR > QR
The angle opposite to the longest side of a triangle is the greatest in size and the angle opposite to the shortest side is the smallest in size. In\(\triangle\)PQR, the longest side is PR and its opposite angle is \(\angle\)Q. The shortest side is PQ and its opposite angle sis R. So, \(\angle\) is the greatest and \(\angle\)R is the smallest angles of \(\triangle\)PQR. Conversely, the side opposite to the greatest angle of a triangle is the longest one and the side opposite to the smallest angle is the shortest one.
The sum of the angles of a triangle is always 180. In\(\triangle\)PQR, \(\angle\)P + \(\angle\)Q + \(\angle\)R = 180
Thhe exterior angle of a triangle is equal to the sum of its two opposite interior angles. In\(\triangle\)PQR, \(\triangle\)PQR, \(\angle\)PRS = \(\angle\)PQR + \(\angle\)QPR
Congruent Triangles
Those triangles which have exactly the same three sides and exactly the same three angles. If the figures have exactly the same shape and size are called congruent figures.
Conditions of congruency of triangles
Two triangles will be congruent under the following conditions
Side-SIde-SIde (S.S.S) axiom
If three sides of one triangle are respectively equal to three corresponding sides of another angle, the triangles are said to be congruent. In\(\triangle\)ABC and\(\triangle\)PQR AB = PQ (S) BC = QR (S) CA = RP (S) \(\therefore\)\(\triangle\)ABC≅\(\triangle\)PQR (S.S.S axiom)
Side-Angle-Side (S.A.S axiom)
If two sides of one triangle are respectively equal to two sides of another triangle and the angle made by them are also equal, the triangle is said to be congruent. In\(\triangle\)ABCand\(\triangle\)PQR AB = PQ (S) \(\angle\)B = \(\angle\)Q (A) BC = QR (S) \(\therefore\)\(\triangle\)ABC≅\(\triangle\)PQR (S.A.S axiom)
Angle-side-Angle (A.S.A axiom)
If two angles of one triangle are respectively equal to two angles of another triangle and the adjacent sides of the angles are also equal, the triangles are said to be congruent. In \(\triangle\)ABC and \(\triangle\)PQR \(\angle\)B = \(\angle\)Q (A) BC = QR (S) \(\angle\)C = \(\angle\)R (A) \(\therefore\) \(\triangle\)ABC≅\(\triangle\)PQR (A.S.A axiom)
Right angle - Hypotenuse - Side (R.H.S axiom)
If hypotenuse and one of two other sides of a right angled triangle are respectively equal to the hypotenuse and a side of other right angled triangle, the triangles are said to be congruent. In Right angles triangles ABC and PQR \(\angle\)B = \(\angle\)Q (R) AC = PR (H) AB = PQ (S) \(\therefore\)\(\triangle\)ABC≅\(\triangle\)PQR (R.H.S axiom)
Similar Triangles
The figures which are exactly the same in the shape but sizes may be different are known as similar figures. In the given figure,\(\triangle\)ABC and\(\triangle\)PQR are similar triangles because they have equal angles and the same shape.
Thus, if three angles of one triangle are respectively equal to three angles of another triangle the are said to be similar.\(\triangle\)ABC is similar to\(\triangle\)PQR is written as\(\triangle\)ABC∼\(\triangle\)PQR. The symbol '∼' is used to denote is similar to.
Here AB and PQ, BC and QR, CA and RP are the corresponding sides of the similar triangle.
The corresponding sides of similar triangles are always proportional, i.e the ratios of the corresponding sides are equal.
The sum of the length of any two sides of a triangle is always greater than the length of its third side. On \(\triangle\)PQR PQ + QR > PR, QR + PR > PQ and PQ + PR > QR
The angle opposite to the longest side of a triangle is the greatest in size and the angle opposite to the shortest side is the smallest in size. In \(\triangle\)PQR, the longest side is PR and its opposite angle is \(\angle\)Q. The shortest side is PQ and its opposite angle sis R. So, \(\angle\) is the greatest and \(\angle\)R is the smallest angles of \(\triangle\)PQR. Conversely, the side opposite to the greatest angle of a triangle is the longest one and the side opposite to the smallest angle is the shortest one.
The sum of the angles of a triangle is always 180. In \(\triangle\)PQR, \(\angle\)P + \(\angle\)Q + \(\angle\)R = 180
Thhe exterior angle of a triangle is equal to the sum of its two opposite interior angles. In \(\triangle\)PQR, \(\triangle\)PQR, \(\angle\)PRS = \(\angle\)PQR + \(\angle\)QPR
It includes every relationship which established among the people.
There can be more than one community in a society. Community smaller than society.
It is a network of social relationships which cannot see or touched.
common interests and common objectives are not necessary for society.