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Triangles

Importance of triangles in construction Importance of triangles in construction There are many practical uses of the triangle, especially in construction work because of its structural support. Many structures use triangular shapes as braces for construction. The roof sections of houses are made of triangular trusses that support the roof and the house.

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°.
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