Properties of Triangles

Exercise 6.2

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

triangles triangles triangles triangles triangles triangles

Answer:

(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:

triangles triangles triangles triangles triangles triangles

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:

triangles triangles triangles triangles triangles triangles

Answer:

  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.

triangles triangles triangles

Answer:

(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°`

triangles triangles triangles

(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°`



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