ICSE Class 8 Maths Geometry Chapter 26 Triangles

Triangles

Elements Of A Triangle

Properties Of Triangles

Terms Associated With Triangles

Congruency Of Triangles

Triangles

Three non-collinear co-planar points A, B, and C joined by three line segments AB, BC, and CA form Triangle ABC.

Elements Of A Triangle

The six elements of a triangle are its three sides and three angles. A, B, C are the vertices of Triangle ABC given below:

Angle 1, Angle 2, Angle 3 are its three interior angles; AB, BC, and CA are its three sides; and Angle 4, Angle 5, Angle 6 are its three exterior angles formed when the sides of the triangle are produced in that order.

Types Of Triangles

According to the magnitude of its interior angles, a triangle may be acute-angled, right-angled, or obtuse-angled.

According to the measure of its sides, a triangle may be equilateral, isosceles, or scalene.

Equilateral - three sides equal

Isosceles - two sides equal

Scalene - no sides equal

Acute-angled - all angles less than 90 degrees

Right-angled - one angle equals 90 degrees

Obtuse-angled - one angle greater than 90 degrees

Properties Of Triangles

1. To prove: The sum of the interior angles of a triangle is 180 degrees.

Proof: Draw Triangle ABC and line l parallel to BC such that l passes through vertex A. Mark Angle 1, Angle 2, Angle 3, Angle 4, and Angle 5 as shown below:

Now Angle 2 equals Angle 4 (alternate interior angles) and Angle 3 equals Angle 5 (alternate interior angles)

But Angle 4 + Angle 5 equals 180 degrees (straight angle)

Therefore Angle 2 + Angle 1 + Angle 3 equals 180 degrees

Thus, the sum of the interior angles of a triangle is 180 degrees. Q.E.D.

Try this! If the sum of two angles of a triangle is 56 degrees, find the third angle. What kind of angle will it be?

Teacher's Note

When building roof trusses or constructing triangular supports, carpenters rely on the fact that all triangles have interior angles summing to 180 degrees to ensure structural integrity and proper fitting of materials.

2. To prove: The sum of the length of two sides is greater than the length of the third side of a triangle.

Proof:

Case I: Construct Triangle ABC, given AB equals 6 cm, BC equals 3 cm, and CA equals 2 cm. The arcs for BC equals 3 cm and CA equals 2 cm do not intersect at any point (Figure 26.1(i)).

Case II: Construct Triangle PQR, given PQ equals 5 cm, QR equals 3 cm, and RP equals 2 cm. The arcs for QR equals 3 cm and RP equals 2 cm do not intersect but meet each other at point R. But point R is on line segment PQ itself (Figure 26.1(ii)).

Thus Triangle ABC and Triangle PQR are not possible with the given measurements or a triangle is possible only if the sum of two sides is greater than the measure of the third side. Q.E.D.

Teacher's Note

When using a triangle of surveying or navigation tools, one must choose three distances that satisfy the triangle inequality - for example, you cannot make a triangle with sticks of lengths 2, 3, and 5 units because 2 plus 3 equals 5, not greater than 5.

3. To prove: The measure of an exterior angle is equal to the sum of its two opposite interior angles.

Proof: Extend side BC of Triangle ABC to form exterior Angle 4.

Now Angle 3 + Angle 4 equals 180 degrees (linear pair)

Therefore Angle 4 equals 180 degrees - Angle 3

But Angle 1 + Angle 2 + Angle 3 equals 180 degrees (from first property)

Therefore Angle 1 + Angle 2 equals 180 degrees - Angle 3

Therefore Angle 1 + Angle 2 equals Angle 4

Or, exterior angle is equal to the sum of opposite interior angles. Q.E.D.

Some Special Terms Related To Triangles

Incentre

The point where the angle bisectors of all three interior angles of a triangle meet is known as its incentre (Figure 26.2(i)).

The incircle (Figure 26.2(ii)), or the circle touching all the three sides of a triangle, is drawn with the incentre as its centre.

Circumcentre

The point where the perpendicular bisectors of all the sides of a triangle meet is known as its circumcentre (Figure 26.3(i)).

The circumcircle (Figure 26.3(ii)), or the circle passing through all the three vertices of a triangle, is drawn with the circumcentre as its centre.

Orthocentre

The point where the three altitudes of a triangle meet is known as its orthocentre. The orthocentre of a scalene triangle lies in its exterior.

Example 1: Find the value of x in Figure 26.7, given ABC is an isosceles triangle.

In the given figure Angle 1 equals 71 degrees (vertically opposite angles)

As the given triangle is an isosceles triangle, Angle 1 equals Angle 2

Therefore Angle 2 equals 71 degrees

x + Angle 1 + Angle 2 equals 180 degrees (sum of interior angles of a triangle)

Therefore x equals 180 degrees - (Angle 1 + Angle 2)

Therefore x equals 180 degrees - (71 degrees + 71 degrees)

Therefore x equals 180 degrees - 142 degrees equals 38 degrees

Median

A median is a line joining the mid-point of a side of a triangle to the vertex opposite it (Figure 26.4).

Centroid

The centroid of a triangle is the point of intersection of its three medians (Figure 26.5).

Altitude

A perpendicular drawn from a vertex to its opposite side is called an altitude of the triangle (Figure 26.6).

AB equals base; CP equals altitude

BC equals base; AQ equals altitude

CA equals base; BR equals altitude

Example 2: Find the value of x in Figure 26.8(i).

Connect points B and C and mark Angle 1 and Angle 2 as shown in Figure 26.8(ii).

In Triangle DBC, Angle 1 + Angle 2 + 124 degrees equals 180 degrees

Angle 1 + Angle 2 equals 180 degrees - 124 degrees equals 56 degrees

In Triangle ABC, x + (27 degrees + Angle 1) + (37 degrees + Angle 2) equals 180 degrees

Therefore x + 27 degrees + 37 degrees + Angle 1 + Angle 2 equals 180 degrees

Therefore x + 27 degrees + 37 degrees + 56 degrees equals 180 degrees (substituting Angle 1 + Angle 2 equals 56 degrees obtained earlier)

Therefore x + 120 degrees equals 180 degrees

Therefore x equals 180 degrees - 120 degrees equals 60 degrees

Teacher's Note

Surveyors use triangulation - dividing a plot of land into triangles and measuring angles - to calculate distances and areas that cannot be directly measured, relying on these angle properties to verify their work.

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