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Introduction

Triangle: A closed figure bounded by three line segments is called a triangle. The line segments forming a triangle are called its 'sides' and each point, where two sides intersect, is called its 'vertex'. We denote a triangle by the symbol Δ. Thus, a Δ PQR has:
(i) three sides, namely PQ, QR and RP;
(ii) three vertices, namely P, Q and R;
(iii) three angles, namely ∠RPQ, ∠PQR and ∠QRP, to be denoted by ∠P, ∠Q and ∠R respectively.


Sometimes lowercase letters are used to denote the length of the sides of a triangle. In the above figure, PQ is the side opposite to the vertex R and is denoted by 'r'. Similarly, the lengths of the sides QR and RP are denoted by 'p' and 'q' respectively because the sides QR and RP are opposite to the vertices P and Q respectively.

Interior and Exterior of a Triangle

Interior of a triangle is the region of the plane enclosed by a triangle. It is shown in gray color in the adjacent figure. The two points X and Y are in the interior of Δ PQR. Interior of a triangle together with the points on the boundary of a triangle is known as the triangular region. In the adjacent figure, X, Y and T are in the triangular region of PQR.

Exterior of a triangle is the region of the plane which lies beyond or not enclosed by the boundary of a triangle. It is shown in blue shade in the adjacent figure. Z is a point which is exterior to the Δ PQR.

Angle Sum Theorem

If the measures of two angles of a triangle are known, how can the measure of the third angle be determined ? The angle sum theorem explains that the sum of the measures of the angles of any triangle is always 180°. In the below figure, in Δ PQR, ∠P + ∠Q + ∠R = 180°.



Ex: If two angles of a triangle are 57° and 62°, then the third angle is
Sol: Let the third angle be x°. We know that the sum of 3 angles of a triangle = 180°.

57° + 62° + x° = 180°
119° + x° = 180°
x = 180° – 119°
= 61°

∴ The third angle = 61°.

Examples of equilateral triangle in real life Examples in real life (Clockwise)
Give way sign; Billiard triangle; Catholic trinity; Traffic signals; Vicks logo; Family planning logo in India.
Equilateral Triangle

A triangle having all sides equal, is called an equilateral triangle. All the angles of an equilateral triangle are equal. Since the sum of all the angles in a triangle is equal to 180°, the measure of each angle of an equilateral triangle is 180/3 or 60°. Triangle ABC shown below is equilateral.

Brain Teaser

Examples of right angled triangle in real life Examples in real life (Clockwise)
Setsquare, View from a ship to light house; Half sandwich; Diameter of a circle subtending 90° on the circumference; Ladder against a wall.

Brain Teaser
Right-angled Triangle

A triangle in which one of the angle measures 90°, is called a right-angled triangle or simply a right triangle. In a right-angled triangle, the side opposite to the right angle is called its hypotenuse and the other two sides are called 'legs'. From the angle sum theorem, it is obvious that a triangle can not have more than one right angle.

In Δ ABC, AC is the hypotenuse and in Δ PQR, PR is the hypotenuse. It is a common practice to represent a right angle as shown in Δ ABC or in Δ PQR without actually mentioning as 90°. You will learn later the relationship between the hypotenuse and the legs of the right-angled triangle given by Pythagoras theorem.

Ex: What are the angles of a right isosceles triangle?
Sol: One of the angles is 90°. Being isosceles, two sides are equal and hence the base angles are also equal. Their sum in this case is 90°. Therefore, each base angle is equal to 45°. So, in any right isosceles triangle the angles are always 90, 45 and 45°. This is shown in Δ PQR above.

You will also learn later the definitions of six trigonometrical ratios and their relationships in a right-angled triangle.