The plane figure bounded by three lines, joining three non collinear points, is called a triangle.

There are various types of triangles with unique properties. However, some properties are applicable to all triangles. One such property is *the sum of any two sides of a triangle is always greater than the third side of the triangle.* It is illustrated as follows:

So *x + y > z, x + z > y*, and *y + z > x.*

**Scalene Triangle:**A triangle in which none of the three sides is equal is called a scalene triangle.**Isosceles Triangle:**A triangle in which at least two sides are equal is called an isosceles triangle.

In an isosceles triangle, the angles opposite to the congruent sides are congruent.

Also, if two angles of a triangle are equal, then the sides opposite to them are also equal.

In ∆ABC, AB = AC, ∠ABC = ∠ACB

**Equilateral Triangle:**Equilateral triangle is a unique triangle in which all the angles & all the sides are equal.

In ∆ABC, AB = BC = AC. Then, ∠ABC = ∠BCA = ∠CAB = 60°

**Acute Triangle:**If all the three angles of a triangle are acute i.e., less than 90°, then the triangle is an acute-angled triangle.**Obtuse Triangle:**If any one of the three angles of a triangle is obtuse (greater than 90°), then that particular triangle is said to be an obtuse angled triangle. Note: the remaining two angles of an obtuse angled triangle are always acute.**Right Triangle:**If any of the three angles of a triangle is a right angle (i.e. exactly 90°), then that particular triangle is know as the right angled triangle.

In the figure above, DABC is a right triangle, so (AB)^{2} + (AC)^{ 2} = (BC)^{2}. Here, AB = 6 and AC= 8, so BC= 10, since 6^{2} + 8^{2} = 36 + 64 = 100 = (BC)^{ 2} and BC = &redic;100.

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Any triangle in which the lengths of the sides are in the ratio 3:4 is always a right angled triangle.

In general, if x, by and z are the lengths of the sides of a triangle in which x^{2} + y^{2} = z^{2}, then the triangle is said to be a right angled triangle.

There are some Pythagorean triplets, which are frequently used in the questions. It is better to memorize these triplets.

- 3, 4 & 5
- 5, 12 & 13
- 7, 24 & 25
- 8, 15 & 17
- 9, 40 & 41
- 11, 60 & 61
- 12, 35 & 37
- 16, 63 & 65
- 20, 21 & 29
- 28, 45 & 53.

Any multiple of these Pythagorean triplets will also be a Pythagorean triplet i.e. when we say is a 5,12, 13triplet, if we multiply all these numbers by 3, it will also be a triplet i.e. 15, 36, 39 will also be a Pythagorean triplet.

For example, in ∆PQR, if PR = 2cm, then PQ = &redic;2cm and QR = &redic;2cm.

In ∆ABC, ∠ABC + ∠BAC + ∠ACB = 180°

In figure on previous page, ∠ABC + ∠ABH = 180°

∠ABC + ∠CBI = 180°

In ∆ABC, AB + BC > AC, also AB + AC > BC and AC + BC > AB.

The area of a triangle is equal to: (the length of the altitude) × (the length of the base) / 2.

BD = 5

In ∆ABC, BD is the altitude to base AC and AE is the altitude to base BC.

The triangle area is also equal to (AE × BC) / 2. If DABC above is isosceles and AB = BC, then altitude BD bisects the base; that is, AD = DC = 4. Similarly, any altitude of an equilateral triangle bisects the side to which it is drawn.

- Two sides & the included angle of a triangle are respectively equal to two sides & included angle of other triangle (SAS).
- 2 angles & 1 side of a triangle are respectively equal to two angles & the corresponding side of the other triangle (AAS).
- Three sides of a triangle are respectively congruent to three sides of the other triangle (SSS).
- 1 side & hypotenuse of a right-triangle are respectively congruent to 1 side & hypotenuse of other rt. triangle (RHS).

Two triangles are said to be similar to each other if they are alike only in shape. The corresponding angles of these triangles are equal but corresponding sides are only proportional. All congruent triangles are similar but all similar triangles are not necessarily congruent.

- Three sides of a triangle are proportional to the three sides of the other triangle (SSS).
- Two angles of a triangle are equal to the two angles of the other triangle (AA) respectively.
- Two sides of a triangle are proportional to two sides of the other triangle & the included angles are equal (SAS).

Suggested Action:

- If two triangles are similar, ratios of sides = ratio of heights = ratio of medians = ratio of angle bisectors = ratio of inradii = ratio of circum radii.
- Ratio of areas = b
_{1}h_{1}/b_{2}h_{2}= (s_{1})^{2}/(s_{2})^{2 }, where b_{1}& h_{1}are the base & height of first triangle and b_{2}& h_{2}are the base & height of second triangle. s_{1}& s_{2}are the corresponding sides of first and second triangle respectively. - The two triangles on each side of the perpendicular drawn from the vertex of the right angle to the largest side i.e. Hypotenuse are similar to each other & also similar to the larger triangle.

∆ DBA is similar to ∆ DCB which is similar to ∆ BCA.

- The altitude from the vertex of the right angle to the hypotenuse is the geometric mean of the segments into which the hypotenuse is divided.

i.e. (DB)^{2} = AD * DC