ICSE Class 7 Maths Chapter 24 Triangles

Chapter 24: Triangles

24.1 Review

Definition Of A Triangle

A closed figure, having 3 sides, is called a triangle and is usually denoted by the Greek letter ∆ (delta).

The figure, given alongside, shows a triangle ABC (∆ ABC) bounded by three sides AB, BC and CA.

Vertex

The point, where any two sides of a triangle meet, is called a vertex.

Clearly, the given triangle has three vertices, namely: A, B and C.

Vertices is the plural of vertex

Interior Angles

In ∆ ABC (given above), the angles BAC, ABC and ACB are called its interior angles as they lie inside the ∆ ABC.

The sum of interior angles of a triangle is always 180°

Exterior Angles

When any side of a triangle is produced the angle so formed, outside the triangle and at its vertex, is called its exterior angle.

For a given triangle ABC, if side BC is produced to the point D, then ∠ACD is its exterior angle. And, if side AC is produced to the point E, then the exterior angle would be ∠BCE.

Thus, at every vertex, two exterior angles can be formed and that these two angles being vertically opposite angles, are always equal.

Also, at each vertex of a triangle, the sum of the exterior angle and its corresponding interior angle is 180°.

In ∆ABC, given alongside,

Exterior angle + Interior angle = 180°

At vertex A: ∠BAE + ∠A = 180°

At vertex B: ∠CBF + ∠B = 180° and

At vertex C: ∠ACD + ∠C = 180°

Interior Opposite Angles

When any side of a triangle is produced, an exterior angle is formed. The two interior angles of this triangle, that are opposite to the exterior angle formed, are called its interior opposite angles.

In the given figure, side BC of ∆ ABC is produced to the point D, so that the exterior ∠ACD is formed. Then the two interior opposite angles are ∠BAC and ∠ABC.

Teacher's Note

Understanding triangle properties helps in construction work, such as determining stable roof angles or calculating land measurements when surveying properties.

Relation Between Exterior Angle And Interior Opposite Angles

Exterior angle of a triangle is always equal to the sum of its two interior opposite angles.

Thus in the figure, given above, ∠ACD = ∠BAC + ∠ABC.

Similarly; in the triangle ABC, drawn alongside,

Exterior angle CAE = ∠B + ∠C

and exterior angle ABF = ∠A + ∠C.

Example 1

(i) Can a triangle have angles 60°, 70° and 70° ?

(ii) Two angles of a triangle are 48° and 73°, find its third angle.

(iii) Three angles of a triangle are (2x + 20)°, (x + 30)° and (2x - 10). Find the angles.

Solution

(i) Since, 60° + 70° + 70° = 200°

A triangle can not have angles 60°, 70° and 70°

Remember: Sum of the angles of a triangle is always 180°

(ii) Sum of two given angles = 48° + 73° = 121°

The third angle = 180° - 121° = 59°

(iii) Since, the sum of the interior angles of a triangle = 180°

∴ (2x + 20) + (x + 30) + (2x - 10) = 180°

⇒ 5x + 40 = 180°

i.e. 5x = 180 - 40 = 140 and x = 140/5 = 28

∴ Required angles = (2x + 20)°, (x + 30)° and (2x - 10)°

= (2 × 28 + 20)°, (28 + 30)° and (2 × 28 - 10)°

= 76°, 58° and 46°

Example 2

Use the figure, given alongside, to find the value of:

(i) x,

(ii) ∠BAC,

(iii) ∠ACB.

Solution

(i) Since, the exterior angle of a ∆ = sum of its two interior opposite angles

∴ 130° = 7x + 6x

⇒ 13x = 130°

⇒ x = 130°/13 = 10°

(ii) ∠BAC = 7x = 7 × 10° = 70°

(iii) ∠ACB = 6x = 6 × 10° = 60°

Teacher's Note

Exterior angle theorem is used in navigation and surveying to calculate angles when direct measurement is not possible, such as finding angles in mountain terrain.

Exercise 24(A)

1. State, if the triangles are possible with the following angles:

(i) 20°, 70° and 90°

(ii) 40°, 130° and 20°

(iii) 60°, 60° and 50°

(iv) 125°, 40° and 15°

2. If the angles of a triangle are equal, find its angles.

3. In a triangle ABC, ∠A = 45° and ∠B = 75°, find ∠C.

4. In a triangle PQR, ∠P = 60° and ∠Q = ∠R, find ∠R.

5. Calculate the unknown marked angles in each figure:

(i) A triangle with angles 90°, x°, and 30°

(ii) A triangle with angles y°, 80°, and 20°

(iii) A right triangle with angles a°, 40°, and a right angle

6. Find the value of each angle in the given figures:

(i) A triangle ABC with angles 5x° at A, 4x° at B, and x° at C

(ii) A triangle ABC with angles x° at A, 2x° at B, and 2x° at C

7. Find the unknown marked angles in the given figures:

(i) A triangle with angles b° and 50°

(ii) A triangle with angles b°, 90°, and x°

(iii) A triangle with angles k°, k°, and k°

(iv) A triangle with angles m° - 5°, 60°, and m° + 5°

8. In the given figure, show that: ∠a = ∠b + ∠c.

(i) If ∠b = 60° and ∠c = 50°, find ∠a.

(ii) If ∠a = 100° and ∠b = 55°, find ∠c.

(iii) If ∠a = 108° and ∠c = 48°, find ∠b.

9. Calculate the angles of a triangle, if they are in the ratio 4 : 5 : 6.

10. One angle of a triangle is 60°. The other two angles are in the ratio of 5 : 7. Find the two angles.

11. One angle of a triangle is 61° and the other two angles are in the ratio 1 1/2 : 1 1/3. Find these angles.

12. Find the unknown marked angles in the given figures.

(i) A triangle with angles x° and 30°, and exterior angle 110°

(ii) A figure showing exterior angles 60° and 120°, with interior angle y°

(iii) A triangle with angles k°, 35°, and exterior angle 122°

(iv) A triangle with angles a°, 73°, and exterior angle 135°

(v) A triangle with angles a°, b°, c°, and exterior angle 140°, with one side 125°

(vi) A triangle with angles 63°, x°, 112°, and exterior angle y°

(vii) A triangle with angles a° and 120°

(viii) A triangle with angle measurements 2m and 4m, and exterior angle 140°

(ix) A triangle with angles b°, a°, and 105°

24.2 Classification Of Triangles

(A) With Regard To Their Angles

1. Acute Angled Triangle

It is a triangle, whose each angle is acute, i.e., each angle is less than 90°.

2. Right Angled Triangle

It is a triangle, whose one angle is a right angle, i.e., equal to 90°.

The figure, given alongside, shows a right angled triangle XYZ as ∠XYZ = 90°.

(i) One angle of a right angled triangle is 90° and the other two angles of it are acute angles, such that their sum is always 90°.

In ∆ xyz, given above, ∠y = 90° and each of ∠x and ∠z is acute such that ∠x + ∠z = 90°.

(ii) In a right angled triangle, the side opposite to the right angle is largest of all its sides and is called the hypotenuse. In given right angled ∆ XYZ, side XZ is the hypotenuse.

3. Obtuse Angled Triangle

If one angle of a triangle is more than 90°, it is called an obtuse angled triangle.

In case of an obtuse angled triangle, each of the other two angles is always acute and their sum is less than 90°.

Teacher's Note

Architects use angle classifications when designing roof structures, ensuring that acute angles provide strength while right angles allow for practical construction methods.

(B) With Regard To Their Sides

1. Scalene Triangle

If all the sides of a triangle are unequal, it is called a scalene triangle.

In a scalene triangle, all its angles are also unequal.

2. Isosceles Triangle

If atleast two sides of a triangle are equal, it is called an isosceles triangle.

In ∆ ABC, shown alongside, side AB = side AC.

∴ ∆ ABC is an isosceles triangle.

(i) The angle contained by equal sides i.e. ∠BAC is called the vertical angle or the angle of vertex.

(ii) The third side (the unequal side) is called the base of the isosceles triangle.

(iii) The two other angles (other than the angle of vertex) are called the base angles of the triangle.

24.3 Important Properties Of An Isosceles Triangle

The base angles, i.e., the angles opposite to equal sides of an isosceles triangle are always equal.

In given triangle ABC,

(i) if side AB = side BC, then angle opposite to AB = angle opposite to BC, i.e., ∠C = ∠A.

(ii) if side BC = side AC, then angle opposite to BC = angle opposite to AC, i.e. ∠A = ∠B and so on.

Conversely: If any two angles of a triangle are equal, the sides opposite to these angles are also equal, i.e., the triangle is isosceles.

Thus in ∆ ABC,

(i) if ∠B = ∠C ⇒ side opposite to ∠B = side opposite to ∠C ⇒ side AC = side AB.

(ii) if ∠A = ∠B ⇒ side BC = side AC and so on.

3. Equilateral Triangle

If all the three sides of a triangle are equal, it is called an equilateral triangle.

In the given figure, ∆ ABC is equilateral, because AB = BC = CA

Also, all the angles of an equilateral triangle are equal to each other and so each angle = 60°

∴ 60° + 60° + 60° = 180°

Since, all the angles of an equilateral triangle are equal, it is also known as equiangular triangle.

An equilateral triangle is always an isosceles triangle, but its converse is not always true.

4. Isosceles Right Angled Triangle

If one angle of an isosceles triangle is 90°, it is called an isosceles right angled triangle.

In the given figure, ∆ ABC is an isosceles right angled triangle, because: ∠ACB = 90° and AC = BC.

Here, the base is AB, the vertex is C and the base angles are ∠BAC and ∠ABC, which are equal.

Since, the sum of the angles of a triangle = 180°

∴ ∠ABC = ∠BAC = 45°

45° + 45° + 90° = 180°

Example 3

In the given isosceles triangle, find the base angles.

Solution

Let each of the base angles be x.

∴ x + x + 40° = 180°

Sum of the angles of a ∆ = 180°

⇒ 2x + 40° = 180°

⇒ 2x = 180° - 40°

⇒ x = 140°/2 = 70°

∴ Each base angle is 70°

Example 4

One base angle of an isosceles triangle is 65°. Find its angle of vertex.

Solution

Since, the base angles of an isosceles triangle are equal,

∴ Other base angle is also 65°.

Let the angle of vertex be x.

∴ x + 65° + 65° = 180°

Sum of the angles of a ∆ = 180°

⇒ x + 130° = 180°

⇒ x = 180° - 130° = 50°

Example 5

If one base angle of an isosceles triangle is double of the vertical angle, find all its angles.

Solution

Draw an isosceles triangle in which mark the vertical angle as x.

∴ The two base angles will be 2x each.

Hence, x + 2x + 2x = 180°

⇒ 5x = 180° and x = 180°/5 = 36°

⇒ 2x = 2 × 36° = 72°

∴ Vertical angle = 36° and each base angle = 72°

Teacher's Note

Isosceles triangles appear in bridge design and tent construction, where equal sides provide balanced weight distribution and structural stability.

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