1. Find the value of the unknown x in the following diagrams:

NOTE: We know that the sum of all the interior angles of a triangle is 180o.
(i)

Solution:
∠BAC + ∠ABC + ∠BCA = 180o
x + 50o + 60o = 180o
x + 110o = 180o
Transposing 110o from LHS to RHS it becomes – 110o
x = 180o – 110o
x = 70o
(ii)

Solution:
The given triangle is a right-angled triangle. So the ∠QPR is 90o.
Now,
∠QPR + ∠PQR + ∠PRQ = 180o
90o + 30o + x = 180o
120o + x = 180o
Transposing 110o from LHS to RHS it becomes – 110o
x = 180o – 120o
x = 60o
(ii)

Solution:
∠XYZ + ∠YXZ + ∠XZY = 180o
110o + 30o + x = 180o
140o + x = 180o
Transposing 140o from LHS to RHS it becomes – 140o
x = 180o – 140o
x = 40o

Solution:
50o + x + x = 180o
50o + 2x = 180o
Transposing 50o from LHS to RHS it becomes – 50o
2x = 180o – 50o
2x = 130o
x = 130o/2
x = 65o
(v)

Solution:
x + x + x = 180o
3x = 180o
x = 180o/3
x = 60o
Hence, the given triangle is an equilateral triangle.
(vi)

Solution:
90o + 2x + x = 180o
90o + 3x = 180o
Transposing 90o from LHS to RHS it becomes – 90o
3x = 180o – 90o
3x = 90o
x = 90o/3
x = 30o
Then,
2x = 2 × 30o = 60o
2.Find the values of the unknowns x and y in the following diagrams:


Solution:
(i)

Solution:
An exterior angle of a triangle is equal to the sum of its interior opposite angles.
Now,
50o + x = 120o
By transposing 50o from LHS to RHS it becomes – 50o
x = 120o – 50o
x = 70o
The sum of all the interior angles of a triangle is 180o.
Now,
50o + x + y = 180o
50o + 70o + y = 180o
120o + y = 180o
Transposing 120o from LHS to RHS it becomes – 120o
y = 180o – 120o
y = 60o
(ii)

Solution:
According to the rule of vertically opposite angles,
y = 80o
Now,
The sum of all the interior angles of a triangle is 180o.
50o + 80o + x = 180o
130o + x = 180o
Transposing 130o from LHS to RHS it becomes – 130o
x = 180o – 130o
x = 50o
(iii)

Solution:
The sum of all the interior angles of a triangle is 180o.
50o + 60o + y = 180o
110o + y = 180o
Transposing 110o from LHS to RHS it becomes – 110o
y = 180o – 110o
y = 70o
Now,
According to the rule of linear pair,
x + y = 180o
x + 70o = 180o
Transposing 70o from LHS to RHS it becomes – 70o
x = 180o – 70o
x = 110o
(iv)

Solution:
According the rule of vertically opposite angles,
x = 60o
The sum of all the interior angles of a triangle is 180o.
30o + x + y = 180o
30o + 60o + y = 180o
90o + y = 180o
Transposing 90o from LHS to RHS it becomes – 90o
y = 180o – 90o
y = 90o
(v)

Solution:
According to the rule of vertically opposite angles,
y = 90o
The sum of all the interior angles of a triangle is 180o.
Now,
x + x + y = 180o
2x + 90o = 180o
Transposing 90o from LHS to RHS it becomes – 90o
2x = 180o – 90o
2x = 90o
x = 90o/2
x = 45o
(vi)

Solution:
According to the rule of vertically opposite angles,
x = y
The sum of all the interior angles of a triangle is 180o.
Now,
x + x + x = 180o
3x = 180o
x = 180o/3
x = 60o
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