ICSE Class 9 Maths Chapter 09 Triangles

Chapter 9: Triangles

Congruency in Triangles

9.1 Introduction

A plane figure bounded by three straight line segments, is called a triangle. Every triangle has three vertices and three sides.

The adjoining figure shows a triangle ABC (\(\triangle ABC\)), whose three

(i) vertices are A, B and C.

(ii) sides are AB, BC and CA.

9.2 Relation Between Sides and Angles of Triangles

1. If all the sides of a triangle are of different lengths, its angles are also of different measures in such a way that, the greater side has greater angle opposite to it.

In the given triangle ABC, sides AB, BC and AC are all of different lengths. Therefore, its angles i.e., \(\angle A\), \(\angle B\) and \(\angle C\) are also of different measures.

Thus, in triangle ABC,

\(AB \neq BC \neq AC\)

\(\Rightarrow \angle A \neq \angle B \neq \angle C\)

Also, according to the given figure, \(AC > BC > AB\)

\(\Rightarrow\) Angle opposite to \(AC\) > angle opposite to \(BC\) > angle opposite to \(AB\)

\(\Rightarrow \angle B > \angle A > \angle C\)

2. Conversely, if all the angles of a triangle have different measures, its sides are also of different lengths in such a way that, the greater angle has greater side opposite to it.

In the given figure,

\(\angle A \neq \angle B \neq \angle C\)

\(\Rightarrow AB \neq BC \neq AC\)

Also, according to the given figure, \(\angle C > \angle A > \angle B\)

\(\Rightarrow\) Side opposite to \(\angle C\) > side opposite to \(\angle A\) > side opposite to \(\angle B\)

\(\Rightarrow AB > BC > AC\)

3. If any two sides of a triangle are equal, the angles opposite to them are also equal. Conversely, if any two angles of a triangle are equal, the sides opposite to them are also equal.

Thus, in triangle ABC,

(i) \(AB = AC \Rightarrow \angle B = \angle C\)

and, (ii) \(\angle B = \angle C \Rightarrow AB = AC\)

4. If all the sides of a triangle are equal, all its angles are also equal. Conversely, if all the angles of a triangle are equal, all its sides are also equal.

Thus, in triangle ABC,

(i) \(AB = BC = AC \Rightarrow \angle A = \angle B = \angle C\)

and, (ii) \(\angle A = \angle B = \angle C \Rightarrow AB = BC = AC\)

Teacher's Note

Understanding that sides and angles relate directly helps students recognize shapes. For instance, if a fence has three equal parts, all angles in that triangular fence will be equal.

9.3 Some Important Terms

1. Median: The median of a triangle, corresponding to any side, is the line joining the mid-point of that side with the opposite vertex.

Examples:

(i) AD is median corresponding to side BC

(ii) BE is median corresponding to side AC

(iii) CF is median corresponding to side AB

1. A triangle has three medians and all the three medians are always concurrent i.e., they intersect each other at one point only.

2. The point of intersection of the medians is called the centroid of the triangle.

In the figure, G is the centroid of \(\triangle ABC\).

3. Also, the centroid of a triangle divides each median in the ratio 2:1.

i.e., in the given figure:

\(AG : GD = 2 : 1; BG : GE = 2 : 1\)

and \(CG : GF = 2 : 1\).

2. Altitude: An altitude of a triangle, corresponding to any side, is the length of the perpendicular drawn from the opposite vertex to that side.

Examples:

(i) AD is altitude corresponding to side BC

(ii) BE is altitude corresponding to side AC

(iii) CF is altitude corresponding to side AB

1. A triangle has three altitudes and all the three altitudes are always concurrent i.e., they intersect each other at one point only.

2. The point of intersection of the altitudes of a triangle is called the orthocentre.

In the given figure, O is the orthocentre of the triangle ABC.

Teacher's Note

Medians and altitudes help us find centers of balance in shapes. The centroid of a triangular piece of cardboard is exactly where it will balance on a pencil point.

Important Properties of Triangles

1. The sum of the angles of a triangle is equal to two right angles i.e. 180°.

2. If one side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

3. As shown, in triangle ABC,

(i) \(\angle A + \angle B + \angle C = 2 \text{ right angles} = 180°\)

(ii) Exterior angle at A = \(\angle B + \angle C\).

(iii) Exterior angle at B = \(\angle A + \angle C\).

(iv) Exterior angle at C = \(\angle A + \angle B\).

Corollaries

Corollary 1: If one side of a triangle is produced, the exterior angle so formed is greater than each of the interior opposite angles.

Corollary 2: A triangle cannot have more than one right angle.

Corollary 3: A triangle cannot have more than one obtuse angle.

Corollary 4: In a right angled triangle, the sum of the other two angles (acute angles) is 90°.

Corollary 5: In every triangle, at least two angles are acute.

Corollary 6: If two angles of a triangle are equal to two angles of any other triangle, each to each, then the third angles of both the triangles are also equal.

Teacher's Note

These properties explain why a triangle's angles always sum to 180 degrees. This is why roof trusses are triangular - they are naturally rigid and stable structures.

9.4 Congruent Triangles

Two triangles are said to be congruent to each other, if on placing one over the other, they exactly coincide.

In fact, two triangles are congruent, if they have exactly the same shape and the same size. i.e., all the angles and all the sides of one triangle are equal to the corresponding angles and the corresponding sides of the other triangle each to each.

Triangles with same shape means: Angles of one triangle are equal to angles of other triangle each to each.

Triangles with same size means: Sides of one triangle are equal to sides of other triangle each to each.

The given figure shows two triangles ABC and DEF such that:

(i) \(\angle A = \angle D\); \(\angle B = \angle E\) and \(\angle C = \angle F\).

(ii) \(AB = DE\); \(BC = EF\) and \(AC = DF\).

\(\therefore \triangle ABC\) is congruent to \(\triangle DEF\)

and we write: \(\triangle ABC \cong \triangle DEF\).

The symbol \(\cong\) is read as "is congruent to".

1. Congruent figures (triangles) always coincide by superposition i.e. by placing one figure over the other.

2. In congruent triangles, the sides and the angles that coincide by superposition are called corresponding sides and corresponding angles.

3. The corresponding sides lie opposite to the equal angles and corresponding angles lie opposite to the equal sides.

In the figure alongside, \(\triangle ABC \cong \triangle EFD\).

Since, \(\angle A = \angle E\), therefore the side opposite to \(\angle A\) and the side opposite to \(\angle E\) are corresponding sides i.e., BC and DF are corresponding sides.

Similarly, AB and EF are corresponding sides as \(\angle C = \angle D\).

Also, AC and DE are corresponding sides.

Conversely, as side \(AB = \) side EF, therefore, angles opposite to these sides i.e. \(\angle C\) and \(\angle D\) are the corresponding angles and so on.

4. Corresponding Parts of Congruent Triangles are also Congruent.

Abbreviated as: C.P.C.T.C.

Teacher's Note

Congruent triangles are used in architecture and construction. For example, identical triangular roof trusses are congruent - they fit perfectly in identical frames.

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