# Properties of Triangles

## Exercise 6.2

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

(i) x = 50° + 70° = 120°
(ii) x = 65° + 45° = 110°
(iii) x = 30° + 40° = 70°
(iv) x = 60° + 60° = 120°
(v) x = 50° + 50° = 100°
(vi) x = 60° + 30° = 90°

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

Answer: Exterior angle of a triangle is equal to the sum of opposite interior angles.

(i) x = 115° - 50° = 65°
(ii) x = 100° - 70° = 30°
(iii) x = 120° - 60° = 60°
(iv) x = 80° - 30° = 50°
(v) x = 75° - 35° = 40°

### Exercise 6.3

Question 1: Find the value of unknown x in following figures:

1. x = 180° - (50° + 60°) = 180° - 110° = 70°
2. x = 180° - (30° + 90°)
Since it is a right angle so the third angle is a right angle.
Or, x = 180° - 120° = 60°
3. x = 180° - (30° + 110°) = 180° - 140° = 40°
4. Here; 50° + 2x = 180°
Or, 2x = 180° - 50° = 130°
Or, x = 130° ÷ 2 = 65°
5. This is an equilateral triangle
Hence, 3x = 180°
Or, x = 180° ÷ 3 = 60°
6. This is a right angled triangle.
Hence, 2x + x + 90°= 180°
Or, 3x = 180° - 90°
Or, x = 90° ÷ 3 = 30°

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

(i) Since an external angle is equal to the sum of opposite exterior angles.
Hence, 120° = 50° + x
Or, x = 120° - 50° = 70°
Now; 120° + y = 180°
Because they make linear pair of angles and angles of a linear pair are always supplementary.
Or, y = 180° - 120° = 60°

(ii) In this case; y = 80°
Because, vertically opposite angles are always equal.
Now, 50° + y + x = 180°
(Angle sum property of triangle)
Or, 50° + 80° + x = 180°
Or, x + 130° = 180°
Or, x = 180° - 130°= 50°

(iii) Here, x = 50° + 60° = 110°
Because , exterior angle in a triangle is equal to sum of opposite internal angles.
Now, x + y = 180° (Linear pair of angles are supplementary)
Or, 110° + y = 180°
Or, y = 180° - 110° = 70°

(iv) Here, x = 60° (Vertically opposite angles are equal)
Now, 30° + x + y = 180° (Angle sum of triangle)
Or, 30° + 60° + y = 180°
Or, y + 90° = 180°
Or, y = 180° - 90°
Or, x = 60° and y = 90°

(v) Here, x = 90° (Vertically opposite angles are equal)
Now, x + x + y = 180°
Or, 2x + 90° = 180°
Or, 2x = 180° - 90° = 90°
Or, x = 90° ÷ 2 = 45°

(vi) Here, x = y (Vertically opposite angles are equal.
Thus, all angles of the given triangle are equal. It means that the given triangle is equilateral triangle and each angle has same measure.
Hence, x = y = 60°