# Triangles

## Triangles

### Congruence of Triangles

We 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.

Two triangles are said to be congruent by using any of the following four rules. They are Side Angle
Side (SAS) Rule, Angle Side Angle (ASA) Rule, Side Side Side (SSS) Rule, Right angle Hypotenuse
Side (RHS) Rule.

### SAS Congruence Rule

If two sides and included angle of one triangle are equal to the corresponding two sides and included
angle of other triangle, then the two triangles are congruent. For two triangles to be congruent, equal
angles must be included between the pairs of equal sides. So, SAS congruence rule holds but not ASS
or SSA rule.

Consider triangles ABC and XYZ, If AB = XY, BC = YZ and ∠B = ∠Y, then ΔABC ≅ ΔXYZ.

### ASA Congruence Rule

If two angles and included side of one triangle are equal to two angles and the included side of other
triangle, then the two triangles are congruent.

Consider triangles ABC and XYZ. If BC = YZ, ∠B = ∠Y and ∠C = ∠Z, then ΔABC ≅ ΔXYZ.

If two pairs of angles of a triangle are equal, then the third pair is also equal since the sum of the three
angles of a triangle is 180°. So, two triangles are congruent if any two pairs of angles and one pair of
corresponding sides are equal. It is called as the AAS Congruence Rule.

### SSS Congruence Rule

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

Consider triangles ABC and XYZ. If AB = XY, BC = YZ and AC = XZ, then ΔABC ≅ ΔXYZ.

### RHS Congruence Rule

Two right angled triangles are said to be congruent if the hypotenuse and one side of one triangle
are equal to the hypotenuse and corresponding side of the other triangle. Note that, the right
angle is not the included angle between the congruent sides. RHS stands for Right angle - Hypotenuse
- Side.

Consider right angled triangles ABC and XYZ. If ∠B = ∠Y = 90°, BC = YZ and AC = XZ, 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.