Class 7 The Triangle and its Properties Ex 6.3

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

(i)

Solution:-

The sum of all the interior angles of a triangle is 180o.

∠BAC + ∠ABC + ∠BCA = 180o

x + 50o + 60o = 180o

x + 110o = 180o

x = 180o – 110o

x = 70o

(ii)

Solution:-

The sum of all the interior angles of a triangle is 180o.

The given triangle is a right-angled triangle. So, the ∠QPR is 90o.

∠QPR + ∠PQR + ∠PRQ = 180o

90o + 30o + x = 180o

120o + x = 180o

x = 180o – 120o

x = 60o

(iii)

Solution:-

The sum of all the interior angles of a triangle is 180o.

∠XYZ + ∠YXZ + ∠XZY = 180o

110o + 30o + x = 180o

140o + x = 180o

x = 180o – 140o

x = 40o

(iv)

Solution:-

The sum of all the interior angles of a triangle is 180o.

50o + x + x = 180o

50o + 2x = 180o

2x = 180o – 50o

2x = 130o

\(x = {130^{\circ}\over3}\)

x = 65o

(v)

Solution:-

The sum of all the interior angles of a triangle is 180o.

x + x + x = 180o

3x = 180o

\(x = {180^{\circ}\over3}\)

x = 60o

∴ the given triangle is an equiangular triangle.

(vi)

Solution:-

The sum of all the interior angles of a triangle is 180o.

90o + 2x + x = 180o

90o + 3x = 180o

3x = 180o – 90o

3x = 90o

\(x = {90^{\circ}\over3}\)

x = 30o

2x = 2 × 30o = 60o

2. Find the values of the unknowns x and y in the following diagrams:

(i)

Solution:-

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

50o + x = 120o

x = 120o – 50o

x = 70o 

We also know that,

The sum of all the interior angles of a triangle is 180o.

50o + x + y = 180o

50o + 70o + y = 180o

120o + y = 180o

y = 180o – 120o 

y = 60o

(ii)

Solution:-

From the rule of vertically opposite angles,

y = 80o

The sum of all the interior angles of a triangle is 180o.

50o + 80o + x = 180o

130o + x = 180o

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

y = 180o – 110o 

y = 70o

From the rule of linear pair,

x + y = 180o

x + 70o = 180o

x = 180o – 70

x = 110o

(iv)

Solution:-

From 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

y = 180o – 90o 

y = 90o

(v)

Solution:-

From the rule of vertically opposite angles,

y = 90o

The sum of all the interior angles of a triangle is 180o.

x + x + y = 180o

2x + 90o = 180o

2x = 180o – 90o 

2x = 90o

\(x = {90^{\circ}\over2}\)

x = 45o

(vi)

Solution:-

From the rule of vertically opposite angles,

 x = y

The sum of all the interior angles of a triangle is 180o.

 x + x + x = 180o

3x = 180o

\(x = {180^{\circ}\over3}\)

 x = 60o