Congruence of TrianglesWe come across many figures having the same shape and same size. Such figures are known as congruent figures.
Two triangles are said to be congruent if all the sides and the angles of one triangle are respectively equal to corresponding sides and angles of other triangle.
Corresponding parts of congruent triangles are equal. This is written in short as 'CPCT'. It is necessary to write the correspondence of vertices in correct order for writing of congruence of triangles in symbolic form.
Consider the triangles ABC and XYZ. If AB = XY, BC = YZ, AC = XZ and ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z, then ΔABC ≅ ΔXYZ.
Properties of Triangles
A triangle is said to be an isosceles triangle if two of its sides are equal. The third side of the triangle is called the base. The median from the vertex to the base is perpendicular to the base and divides the triangle into two congruent right angled triangles.
Angles opposite to equal sides of an isosceles triangle are equal.
In ΔABC, if the sides AB = AC, then ∠B = ∠C.
The sides opposite to equal angles of a triangle are equal.
In ΔABC, if the angles ∠B = ∠C, then AB = AC.
The three angles of an equilateral triangle are equal.
Inequalities in a Triangle
A triangle has six parts namely three sides and three angles. If two sides of a triangle are equal then the angles opposite to them are also equal and vice versa. In a triangle the angles and the lengths of the sides are proportional.
If two sides of a triangle are unequal, the angle opposite to the longer side is larger.
In any triangle, the side opposite to the larger angle is longer.
The sum of any two sides of a triangle is greater than the third side.
If a, b and c are three sides of a triangle, then a + b > c, b + c > a, c + a > b.