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

(a) Answer: `x = 180° - (50° + 60°) ``= 180° - 110° = 70°`

triangles

(b) Answer: `x = 180° - (30° + 90°)`

Since it is a right angle so the third angle is a right angle.

Or, `x = 180° - 120° = 60°`

triangles

(c) Answer: `x = 180° - (30° + 110°) ``= 180° - 140° = 40°`

triangles

(d) Answer: Here; `50° + 2x = 180°`

Or, `2x = 180° - 50° = 130°`

Or, `x = 130° ÷ 2 = 65°`

triangles

(e) Answer: This is an equilateral triangle

Hence, `3x = 180°`

Or, `x = 180° ÷ 3 = 60°`

triangles

(f) Answer: 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

(i) Answer: 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°`

triangles

(ii) Answer: 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°`

triangles

(iii) Answer: 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

(iv) Answer: 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°`

triangles

(v) Answer: 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°`

triangles

(vi) Answer: 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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