GSEB Class 7 Maths Solutions Chapter 6 The Triangles and Its Properties Exercise 6.2

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

Answer:
(i) The internal opposite angles are 50° and 70°.
\( \therefore \) Applying the exterior angle property of a triangle, we get that the exterior angle equals the sum of its interior opposite angles.
So, \( x = 50° + 70° = 120° \).
(ii) The interior opposite angles measure 65° and 45°.
\( \therefore \) Using the relationship, the exterior angle is the sum of the interior opposite angles.
We find \( x = 65° + 45° = 110° \).
(iii) The interior opposite angles are 30° and 40°.
\( \therefore \) Applying this relation, the exterior angle is calculated by adding the interior opposite angles.
We get \( x = 30° + 40° = 70° \).
(iv) The interior opposite angles are 60° and 60°.
\( \therefore \) Using the same principle, the exterior angle is simply the sum of the internal opposite angles.
Thus, \( x = 60° + 60° = 120° \).
(v) The interior opposite angles are 50° and 50°.
\( \therefore \) Following this rule, the exterior angle is obtained by summing the two interior opposite angles.
Therefore, \( x = 50° + 50° = 100° \).
(vi) The interior opposite angles are 30° and 60°.
So, \( x = 30° + 60° = 90° \).

Exam Tip: Remember that the exterior angle of any triangle is always equal to the total of its two opposite interior angles. This rule helps solve many problems quickly.

Question 2. Find the value of the unknown interior angle x in the following figures:

Answer:
(i) The exterior angle is 115°. One of the interior opposite angles is 50°.
Applying the relationship: [Sum of the interior opposite angles] = [Exterior angle].
We have \( x + 50° = 115° \).
\( \implies x = 115° - 50° \).
\( \implies x = 65° \).
Therefore, the measure of the unknown interior angle is 65°.
(ii) The exterior angle is 100°. The interior opposite angles are 70° and \( x \).
Using the relation: [Sum of the interior opposite angles] = [Exterior angle].
We get \( 70° + x = 100° \).
\( \implies x = 100° - 70° \).
\( \implies x = 30° \).
Hence, the required measure of the unknown interior opposite angle is 30°.
(iii) The exterior angle is 125°. The interior opposite angles are 90° and \( x \).
Applying the relation: [Sum of the interior opposite angles] = [Exterior angle].
We find \( x + 90° = 125° \).
\( \implies x = 125° - 90° \).
\( \implies x = 35° \).
Thus, the value of the unknown interior opposite angle is 35°.
(iv) The exterior angle is 120°. The interior opposite angles are 60° and \( x \).
Using the relation: [Sum of the interior opposite angles] = [Exterior angle].
We have \( x + 60° = 120° \).
\( \implies x = 120° - 60° \).
\( \implies x = 60° \).
Consequently, the measure of the unknown interior opposite angle is 60°.
(v) The exterior angle is 80°. The interior opposite angles are 30° and \( x \).
Applying the relation: [Sum of the interior opposite angles] = [Exterior angle].
We get: \( 30° + x = 80° \).
\( \implies x = 80° - 30° \).
\( \implies x = 50° \).
Hence, the measure of the unknown interior angle is 50°.
(vi) The exterior angle is 75°. The interior opposite angles are \( x \) and 35°.
Using the relation: [Sum of the interior opposite angles] = [Exterior angle].
We have \( x + 35° = 75° \).
\( \implies x = 75° - 35° \).
\( \implies x = 40° \).
Therefore, the measure of the interior opposite angle is 40°.

Exam Tip: When finding an unknown interior angle, rearrange the exterior angle property formula to subtract the known interior angle from the exterior angle. Always double-check your subtraction.