NCERT Book Class 7 Maths The Triangle and Its Properties

The Triangle and its Properties

6.1 INTRODUCTION

A triangle, you have seen, is a simple closed curve made of three line segments. It has three vertices, three sides and three angles. Here is ΔABC (Fig 6.1). It has

Sides: AB, BC , CA

Angles: ∠BAC, ∠ABC, ∠BCA

Vertices: A, B, C

The side opposite to the vertex A is BC. Can you name the angle opposite to the side AB?

You know how to classify triangles based on the (i) sides (ii) angles.

(i) Based on Sides: Scalene, Isosceles and Equilateral triangles.

(ii) Based on Angles: Acute-angled, Obtuse-angled and Right-angled triangles.

Make paper-cut models of the above triangular shapes. Compare your models with those of your friends and discuss about them.

6.2 MEDIANS OF A TRIANGLE

Given a line segment, you know how to find its perpendicular bisector by paper folding. Cut out a triangle ABC from a piece of paper (Fig 6.3). Consider any one of its sides, say, BC . By paper-folding, locate the perpendicular bisector of BC . The folded crease meets BC at D, its mid-point. Join AD. Fig 6.3

The line segment AD, joining the mid-point of BC to its opposite vertex A is called a median of the triangle. Consider the sides AB and CA and find two more medians of the triangle. A median connects a vertex of a triangle to the mid-point of the opposite side.

6.3 ALTITUDES OF A TRIANGLE

Make a triangular shaped cardboard ABC. Place it upright, on a table. How “tall” is the triangle? The height is the distance from vertex A (in the Fig 6.4) to the base BC .

From A to BC you can think of many line segments (see the next Fig 6.5). Which among them will represent its height? The height is given by the line segment that starts from A, comes straight down to BC , and is perpendicular to BC .

This line segment AL is an altitude of the triangle. An altitude has one end point at a vertex of the triangle and the other on the line containing the opposite side. Through each vertex, an altitude can be drawn.

6.4 EXTERIOR ANGLE OF A TRIANGLE AND ITS PROPERTY

You may repeat the above two activities by drawing some more triangles along with their exterior angles. Every time, you will find that the exterior angle of a triangle is equal to the sum of its two interior opposite angles. A logical step-by-step argument can further confirm this fact.

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

Given Consider ΔABC.

∠ACD is an exterior angle.

To Show: m∠ACD = m∠A + m∠B

Through C draw CE, parallel to BA .

Justification Steps Reasons

(a) ∠1 = ∠x BA || CE and AC is a transversal.

Therefore, alternate angles should be equal.

(b) ∠2 = ∠y BA || CE and BD is a transversal.

Therefore, corresponding angles should be equal.

(c) ∠1 + ∠2 = ∠x + ∠y

(d) Now, ∠x + ∠y = m ∠ACD From Fig 6.9

Hence, ∠1 + ∠2 = ∠ACD

The above relation between an exterior angle and its two interior opposite angles is referred to as the Exterior Angle Property of a triangle.

6.5 ANGLE SUM PROPERTY OF A TRIANGLE

There is a remarkable property connecting the three angles of a triangle. You are going to see this through the following four activities.

1. Draw a triangle. Cut on the three angles. Rearrange them as shown in Fig 6.13 . The three angles now constitute one angle. This angle is a straight angle and so has measure 180°. Fig 6.13

Thus, the sum of the measures of the three angles of a triangle is 180°.

2. The same fact you can observe in a different way also. Take three copies of any triangle, say ΔABC (Fig 6.14).

Arrange them as in Fig 6.15.

What do you observe about ∠1 + ∠2 + ∠3?

(Do you also see the ‘exterior angle property’?)

3. Take a piece of paper and cut out a triangle, say, ΔABC (Fig 6.16).

Make the altitude AM by folding ΔABC such that it passes through A.

Fold now the three corners such that all the three vertices A, B and C touch at M. You find that all the three angles form together a straight angle. This again shows that the sum of the measures of the three angles of a triangle is 180°.

4. Draw any three triangles, say ΔABC, ΔPQR and ΔXYZ in your notebook. Use your protractor and measure each of the angles of these triangles.

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