ICSE Class 8 Maths Chapter 24 Triangles

Chapter 24: Triangles

24.1 Review

Triangle

A triangle is a plane and closed figure, bounded by three straight line segments.

The figure alongside shows a triangle ABC. Symbolically triangle ABC is written as \(\triangle ABC\), where the symbol \(\triangle\) is read as 'triangle'.

(i) The three straight line segments forming a triangle are called sides of the triangle. Thus, AB, BC and CA are the three sides of \(\triangle ABC\).

(ii) The point at which two sides of a triangle meet is called vertex of the triangle. Thus, \(\triangle ABC\) has three vertices A, B and C.

Vertices is the plural of vertex.

Types of Triangles

(a) According to their sides:

(i) Equilateral Triangle - All the sides are equal. Each angle is 60 degrees.

(ii) Isosceles Triangle - Two sides are equal. Angles opposite to equal sides are equal.

(iii) Scalene Triangle - No sides equal. No angles equal.

(b) According to their angles:

(i) Right-angled Triangle - One angle must be 90 degrees.

(ii) Obtuse-angled Triangle - One angle must be obtuse, i.e. greater than 90 degrees and less than 180 degrees.

(iii) Acute-angled Triangle - Each angle is acute, i.e. less than 90 degrees.

1. Axioms

The self evident truths which are accepted without any proof are called axioms. E.g.

(i) If x is greater than y, then y is less than x.

(ii) If a and b both are equal to c, then a = b.

2. Theorem

A proposition that requires proof is called a theorem.

3. Proof

The course of reasoning which establishes the truth or falsity of a statement is called a proof.

4. Corollary

A proposition, whose truth can easily be deducted from a preceding theorem is called its corollary.

24.2 Proving a Theorem

Giving the general enunciation, set out the work in the following order:

1. Draw the figure.

2. Using the letters of your figure, state what is given.

3. State what is required to be proved.

4. State the construction, if necessary.

5. State the proof giving the statement and reasons separately, using suitable abbreviated references.

Theorem 1

The sum of the angles of a triangle is equal to two right angles. (i.e. 180 degrees).

Given:

A triangle ABC.

To Prove:

\(\angle BAC + \angle ABC + \angle BCA = 180°\)

i.e. \(\angle A + \angle B + \angle C = 180°\)

Construction:

Produce the side BC upto point D and draw CE parallel to BA.

Proof:

Hence Proved.

Alternative Method:

Construction:

Through vertex A of the \(\triangle ABC\), draw DE parallel to base BC.

Proof:

Hence Proved.

Example 1:

In triangle ABC; \(6\angle A = 4\angle B = 3\angle C\). Find each angle of the triangle.

Solution:

Let \(6\angle A = 4\angle B = 3\angle C = x\)

\(\Rightarrow \angle A = \frac{x}{6}\), \(\angle B = \frac{x}{4}\) and \(\angle C = \frac{x}{3}\)

\(\Rightarrow \frac{x}{6} + \frac{x}{4} + \frac{x}{3} = 180°\)

\([\because \angle A + \angle B + \angle C = 180°]\)

On solving, we get: \(x = 240°\)

\(\therefore \angle A = \frac{x}{6} = \frac{240°}{6} = 40°\), \(\angle B = \frac{x}{4} = \frac{240°}{4} = 60°\)

and, \(\angle C = \frac{x}{3} = \frac{240°}{3} = 80°\)

(Ans.)

Teacher's Note

Understanding angle relationships in triangles helps students solve real-world problems involving structural design and engineering, where triangular frameworks are fundamental.

Test Yourself

1. If two angles of a triangle are 52 degrees and 88 degrees, its third angle = ..................

2. Two angles of a triangle are 75 degrees each, its third angle = ..................

3. Two angles of a triangle are equal and its third angle is 80 degrees, then each of the equal angles is ..................

4. Angles of a triangle are in the ratio 4:5:3. The largest angle of the triangle = ......................... = ......... and its smallest angle is ......................... = ...........

5. The angles of a triangle are in the ratio 3:7:5; the largest angle = ......................... = .............. and the sum of its other two angles = ................................................. and the special name of this triangles is ..........................

Exercise 24 (A)

1. The angles of a triangle are (3x) degrees, (2x - 7) degrees and (4x - 11) degrees. Find the value of x and measure of each angle of the triangle.

2. One angle of a triangle is 78 degrees and the other two angles are in the ratio 7:10. Calculate the two unknown angles of the triangle.

3. In a triangle ABC, \(\angle A = 2\angle B = 3\angle C\). Find each angle of the triangle.

4. Use the informations given in the figure below to calculate the values of x and y.

5. (i) In \(\triangle ABC\), \(\angle A = 2x + 15°\), \(\angle B = 3x - 5°\) and \(\angle C = 4x + 35°\). Find each angle of the triangle.

(ii) In \(\triangle ABC\), \(\angle A = x + 15°\), \(\angle B = x\) and \(\angle C = 2x - 35°\). Find each angle of the triangle and then assign a special name to the triangle.

(iii) In \(\triangle ABC\), \(\angle A = x + 20°\), \(\angle B = 2(x - 10°)\) and \(\angle C = \frac{3}{2}x\). Show that the triangle is equilateral.

6. From the figure, given below, find the value of \(\angle ABC + \angle BCD + \angle CDE\).

7. In triangle ABC, the bisectors of angles B and C meet at P. Prove that: \(\angle BPC = 90° + \frac{1}{2}\angle A\).

8. The adjoining figure, shows a triangle ABC in which \(\angle A = 58°\), \(\angle B = 60°\) and bisectors of angles B and C meet at O. Find the measure of the angle BOC.

9. In triangle ABC, \(\angle B = 3\angle A - 25°\) and \(\angle A = \angle C\). Find angles A and B.

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