Chapter 9.3 - Equilateral and Isosceles Triangles

In the previous section we saw Exterior angle. In this section, we will discuss about the sum of all the 3 interior angles of a triangle.

Equilateral triangle

An equilateral triangle has the following properties:
• All the 3 sides of an equilateral triangle are of the same length
• All the 3 angles of an equilateral triangle are the same, and is equal to 60
When all the angles are equal, each angle will be equal to 180/3 = 60 (∵ the sum of interior angles of any triangle is equal to 180)
The fig.9.12(a) shows an example of an equilateral triangle:
The '=' sign shown on the sides indicates that the lengths are equal.

Fig.9.12

Isosceles triangle

An isosceles triangle has the following properties:
• Two sides of an isosceles triangle are equal. These two sides are called the equal sides of the isosceles triangle.
• The third side which have a different length from that of the equal sides is called the base
• The angles opposite to the equal sides are called base angles
• The two base angles are equal
Fig.9.12(b) shows an example of an isosceles triangle. 
• PR and QR are the equal sides
• PQ is the base
• ∠RPQ and ∠PQR are the base angles

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We will now see some solved examples:
Solved example 9.5
Find the value of x in the following figs

Fig.9.13

Solution:
(a) • Two sides are shown to be equal. So it is an isosceles triangle.
• The angles opposite to these equal sides are the base angles 35 and x
• In an isosceles triangle, the base angles are equal
∴ x = 35^o
(b)  • Two sides are shown to be equal. So it is an isosceles triangle.
• The angles opposite to these equal sides are the base angles
• In an isosceles triangle, the base angles are equal
• So the two base angles are 45^o
• The sum of all interior angles of a triangle is 180^o
• So we can write: 45 + 45 + x = 180 ⇒90 + x =180
∴ x = 180 - 90 = 90^o
(c) • Two sides are shown to be equal. So it is an isosceles triangle.
• The angles opposite to these equal sides are the base angles
• In an isosceles triangle, the base angles are equal
• So the two base angles are x
• The sum of all interior angles of a triangle is 180^o
• So we can write: x + x + 30 = 180 ⇒2x + 30 =180 ⇒ 2x = 180 - 30 = 150
∴ x = 150/2 = 75^o  
(d) • Two sides are shown to be equal. So it is an isosceles triangle.
• The angles opposite to these equal sides are the base angles
• In an isosceles triangle, the base angles are equal
• So the two base angles are x
• The sum of all interior angles of a triangle is 180^o
• So we can write: x + x + (180 -100) = 180 (∵ 100 and the third interior angle form a linear pair)
• So 2x + 80 =180 ⇒ 2x = 180 - 80 = 100
• x = 100/2 = 50^o
Solved example 9.6
Find the value of x and y in the following figs:

Fig.9.14

Solution:
(a) • Two sides are shown to be equal. So it is an isosceles triangle.
• The angles opposite to these equal sides are the base angles
• In an isosceles triangle, the base angles are equal
• One base angle is x. The other is the opposite angle of 42. So it is also 42
• Thus we get x = 42^o
• The sum of all interior angles of a triangle is 180^o
• So we can write: y + 42 + 42 = 180 ⇒y + 84 = 180
∴ y = 180 -84 = 96^o  
(b) • Two sides are shown to be equal. So it is an isosceles triangle.
• The angles opposite to these equal sides are the base angles
• In an isosceles triangle, the base angles are equal
• So the two base angles are x
• The sum of all interior angles of a triangle is 180^o
• So we can write: x + x + 110 = 180 (∵ The third angle is the opposite angle of 110, which is 110 itself)
• So 2x + 110 =180 ⇒ 2x = 180 - 110 = 70
• x = 70/2 = 35^o
• Now, y forms a linear pair with x. 
• So y = 180 - x ⇒180 -35 = 145^o

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In the next section we will discuss about the 'Sum of two sides of a Triangle'.

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