ICSE Class 8 Maths Geometry Chapter 02 Triangles

Triangles

About Triangles

A closed plane figure bounded by three line segments is called a triangle. In the figure, ABC is a triangle bounded by three line segments AB, BC and CA. These are called the sides of the triangle. The points A, B and C are called the vertices of the triangle. The angles ∠BAC, ∠ABC and ∠BCA are called the interior angles or simply, the angles of the triangle. The three sides and the three angles of a triangle are together called the six parts (or elements) of the triangle.

Exterior Angles

If the side BC of ∠ABC is produced to the point O then ∠ACO is called an exterior angle of ∠ABC at C. ∠ACB and ∠ACO are adjacent angles, so, ∠ACB + ∠ACO = 180°.

Exterior angle + adjacent interior angle = 180°

If AC is produced to D then ∠BCD will be another exterior angle of the ∠ABC at C. Being vertically opposite angles, ∠BCD = ∠ACO. Similarly, there are two exterior angles of equal magnitude at each vertex.

Classification Of Triangles On The Basis Of Angles

Acute-angled

In an acute-angled or acute triangle, each of the angles is less than 90°.

Obtuse-angled

In an obtuse-angled or obtuse triangle, one of the angles is greater than 90°.

Right-angled

In a right-angled or right triangle, one of the angles is a right angle. The side opposite to it is called the hypotenuse.

Classification Of Triangles On The Basis Of Sides

Scalene Triangle

No two sides of a scalene triangle are equal. In the figure, ABC is a scalene triangle as AB ≠ BC ≠ CA.

Isosceles Triangle

An isosceles triangle has two equal sides. In the figure, ABC is an isosceles triangle in which AB = AC (equal sides are marked by an equal number of strokes). The third side BC is called the base of the triangle, while ∠ABC and ∠ACB are called the base angles. ∠BAC is called the vertical angle.

Property: The angles opposite to the equal sides of an isosceles triangle are equal.

In the figure, AB = AC. So, ∠ABC = ∠ACB.

Converse: The converse, or opposite, of this is also true. Thus, if two angles of a triangle are equal, the sides opposite to them are equal.

In the figure, ∠XYZ = ∠XZY. So, XZ = XY. This also implies that the angles of a scalene triangle are all unequal.

Equilateral Triangle

All the three sides of an equilateral triangle are equal. In the adjoining figure, ABC is an equilateral triangle as AB = BC = CA.

Property: All the angles of an equilateral triangle are equal.

In the figure, ∠BAC = ∠ABC = ∠ACB.

Converse: If all the angles of a triangle are equal, it must be an equilateral triangle.

In the adjoining figure, ∠P = ∠Q = ∠R. Hence, PQR is an equilateral triangle.

Angle Properties Of A Triangle

Property 1: The sum of the three angles of a triangle is 180°.

Take a triangle ABC. Draw the line DE parallel to BC through A.

Then ∠ABC = alternate ∠DAB ... (1) and ∠ACB = alternate ∠EAC ... (2) Adding (1) and (2), ∠ABC + ∠ACB = ∠DAB + ∠EAC. Adding ∠BAC to both sides of the above, ∠BAC + ∠ABC + ∠ACB = ∠DAB + ∠BAC + ∠EAC = a straight angle = 180°. Hence, ∠BAC + ∠ABC + ∠ACB = 180°.

Example

In a ∠XYZ, ∠X = 45° and ∠Y = 75°. Find ∠Z.

Solution: In ∠XYZ, ∠X + ∠Y + ∠Z = 180° or 45° + 75° + Z = 180° or ∠Z = 180° - 120° = 60°.

Example

Prove that each angle of an equilateral triangle is 60°.

Solution: In the figure, ABC in an equilateral triangle. ∴ AB = BC = CA ⇒ ∠A = ∠B = ∠C. But ∠A + ∠B + ∠C = 180 ⇒ ∠A + ∠A + ∠A = 180° ⇒ 3∠A = 180° ⇒ ∠A = 180°/3 = 60°. Hence, ∠A = ∠B = ∠C = 60°.

Property 2: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Take ∠ABC. Produce BC to D and through C, draw CE || BA. Then ∠A = alternate ∠ACE and ∠B = corresponding ∠ECD. But, ∠ACE + ∠ECD = ∠ACD. So, ∠ACD = ∠A + ∠B.

Example

Find ∠BAC from the figure.

Solution: ∠BCD = ∠BAC + ∠ABC ⇒ 125° = ∠BAC + 65° ⇒ ∠BAC = 125° - 65° = 60°.

Some Important Terms

Altitude

An altitude of a triangle is the perpendicular drawn from any vertex of the triangle to the opposite side. A triangle has three altitudes. In the figure, AL, BM and CN are the altitudes of ∠ABC.

The three altitudes of a triangle pass through a common point called the 'orthocentre' of the triangle.

Median

The line segment joining a vertex of a triangle to the mid-point of the opposite side is called a median of the triangle. A triangle has three medians. In the figure, L, M and N are the mid-points of the sides BC, CA and AB respectively. So, AL, BM and CN are the medians of ∠ABC.

The three medians of a triangle intersect at a point called the centroid or centre of gravity of the triangle.

Incircle

The circle that lies inside a triangle and touches its three sides is called the incircle. The centre of the incircle is called the incentre. The incentre is the point at which the three internal bisectors of the angles of the triangle meet. In the figure, AI, BI and CI bisect ∠A, ∠B and ∠C respectively and meet at I, which is the incentre.

Circumcircle

The circumcircle of a triangle is the circle that passes through its three vertices. Its centre is called the circumcentre. The point at which the perpendicular bisectors of the sides of the triangle meet is the circumcentre.

Solved Examples

Example 1

Find x from the figure.

Solution: The sum of the angles of a triangle = 180°. ∴ in ∠OBD, ∠BOD + 60° + 38° = 180° or ∠BOD = 180° - 98° = 82°. ∴ ∠AOC = vertically opposite ∠BOD = 82°. Now, in ∠AOC, x + 53° + 82° = 180° or x = 180° - 135° = 45°.

Example 2

Find x and y from the adjoining figure.

Solution: In ∠ABC, AB = AC ⇒ ∠ABC = ∠ACB = x. Also, ∠BAC + ∠ABC + ∠ACB = 180° ⇒ 58° + x + x = 180° ⇒ 2x = 180° - 58° = 122° ⇒ x = 61°.

Also, ∠ACD = 180° - x = 180° - 61° = 119° Now, in ∠ACD, ∠CAD + ∠ACD + ∠ADC = 180°. ⇒ 36° + 119° + y = 180° ⇒ y = 180° - 155° = 25°. Thus, x = 61° and y = 25°.

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