Part A · FoundationPrerequisite · nothing assumed

Triangle Basics ത്രികോണം

Before we can talk about similar triangles we need to be sure about what a triangle is, how we name its parts, and the few facts every triangle obeys. Everything on this page is used later in the chapter, so take your time and play with each figure.

1. What is a triangle?

A triangle (ത്രികോണം) is a closed shape made of three straight sides. The three corners are called vertices (മൂലകൾ; one corner is a vertex), the three line segments are the sides (വശങ്ങൾ) and at every corner the two sides make an angle (കോൺ).

We name a triangle by its three vertices: ΔABC has vertices A, B, C, sides AB, BC, CA and angles ∠A, ∠B, ∠C.

🖐️ Which side is opposite an angle?

Click a button (or click near a corner of the triangle).

The side opposite an angle is the one side that does not touch that corner. This idea is used on every page of the chapter, so make sure it feels automatic.

🌍 Triangles everywhere

Look at a roof truss, the frame of a bicycle, the Howrah bridge, a slice of samosa, the set square in your geometry box. Engineers love triangles because a triangle with fixed side lengths cannot change its shape – it is rigid. (A rectangle can collapse into a parallelogram; a triangle cannot.)

2. Types of triangles

We sort triangles in two ways – by their sides and by their angles.

🖐️ Drag the corners – what kind of triangle is it now?

Lengths are shown in "cm" (40 px = 1 cm). Try to make a right triangle by hand – the readout tells you when an angle is 90°.

3. The angles of a triangle add up to 180°

This is the single most useful fact about triangles. Let us see it first, then prove it.

🎬 Tear off the corners

🎬 The proof with a parallel line

⭐ Angle-sum property

In every triangle, ∠A + ∠B + ∠C = 180°. So if you know two angles you can always find the third: third angle = 180° − (sum of the other two).

4. The exterior angle

Extend one side of a triangle past a corner. The angle formed outside is called an exterior angle (ബാഹ്യകോൺ). It equals the sum of the two interior angles far away from it. (Why? The exterior angle and the interior angle at that corner make 180°, and so do the three interior angles.)

🖐️ Drag A and watch the exterior angle at C

5. Big angle, big side

In any triangle the largest angle faces the longest side and the smallest angle faces the shortest side. Think of a door: open it wider (bigger angle) and the gap between the door and the frame (the opposite side) grows.

🖐️ Check it by dragging

6. Can these three sticks make a triangle? (Triangle inequality)

Three sticks make a triangle only if every two of them together are longer than the third. If two short sticks cannot reach across the long one, the triangle never closes.

🎚️ Choose three lengths and let the sticks swing up

a b c (base)

7. Perimeter and area

The perimeter (ചുറ്റളവ്) is the total length of the boundary: AB + BC + CA. If the sides are 3 cm, 4 cm and 5 cm the perimeter is 12 cm – the length of string needed to go once around.

The area (പരപ്പളവ്) is the space inside. For a triangle:

Area = 12 × base × height

The height (ഉയരം, also called the altitude, ലംബം) is the perpendicular distance from the opposite vertex to the base – measured straight down, not along a slanted side.

🎬 Why "half of base × height"?

8. Three special lines in a triangle

From any vertex you can draw three different useful lines to the opposite side. All three appear in this chapter.

9. Right triangles and Pythagoras

A right triangle (മട്ടത്രികോണം) has one angle of exactly 90°. The longest side, opposite the right angle, is the hypotenuse (കർണ്ണം). The other two sides are the perpendicular sides or legs (ലംബവശങ്ങൾ).

🎬 Pythagoras: the squares on the sides

⭐ Pythagoras' theorem

In a right triangle with perpendicular sides a, b and hypotenuse c: a² + b² = c². Famous examples: 3-4-5, 5-12-13, 6-8-10, 8-15-17. If you know any two sides of a right triangle, you can find the third.

🌍 Real life

Masons check that a wall corner is a true right angle by measuring 3 feet along one wall, 4 feet along the other, and checking that the diagonal is exactly 5 feet. Carpenters, tilers and surveyors all use "3-4-5" every day.

10. Check yourself

🧭 Summary

ΔABC: vertices A, B, C; side opposite ∠A is BC (the side that does not touch A).
By sides: equilateral (3 equal), isosceles (2 equal), scalene (none). By angles: acute, right (one 90°), obtuse (one > 90°).
Angle sum is 180°; an exterior angle equals the sum of the two far interior angles.
Largest angle ↔ longest side. Any two sides together are longer than the third.
Area = ½ × base × height; height is measured perpendicular to the base.
Altitude, median and angle bisector are three different lines from a vertex.
Right triangle: a² + b² = c² (hypotenuse c).

Next: Congruent (equal) triangles – when are two triangles exactly the same? ▶