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

(i) $$\angle$$x + 50° = 120° (Exterior angle of a triangle)
$$\therefore$$ $$\angle$$x = 120°- 50° = 70°
$$\angle$$x + $$\angle$$y + 50° = 180° (Angle sum property of a triangle)
70° + $$\angle$$y + 50° = 180°
$$\angle$$y + 120° = 180°
$$\angle$$y = 180° – 120°
$$\therefore$$ $$\angle$$y = 60°
Thus $$\angle$$x = 70 and $$\angle$$y – 60°

(ii) $$\angle$$y = 80° (Vertically opposite angles are same)
$$\angle$$x + $$\angle$$y + 50° = 180° (Angle sum property of a triangle)
$$\therefore$$ $$\angle$$x + 80° + 50° = 180°
$$\therefore$$ $$\angle$$x + 130° = 180°
$$\therefore$$$$\angle$$x = 180° – 130° = 50°
Thus, $$\angle$$x = 50° and $$\angle$$y = 80°

(iii) $$\angle$$y + 50° + 60° = 180° (Angle sum property of a triangle)
$$\angle$$y + 110° = 180°
$$\therefore$$$$\angle$$y = 180°- 110° = 70°
$$\angle$$x + $$\angle$$y = 180° (Linear pairs)
Or, $$\angle$$x + 70° = 180°
$$\therefore$$ $$\angle$$x = 180° – 70° = 110°
Thus, $$\angle$$x = 110° and y = 70°

(iv) $$\angle$$x = 60° (Vertically opposite angles)
$$\angle$$x + $$\angle$$y + 30° = 180° (Angle sum Or, 60° + $$\angle$$y + 30° = 180°
Or, $$\angle$$y + 90° = 180°
Or, $$\angle$$y = 180° – 90° = 90°
Thus, $$\angle$$x = 60° and $$\angle$$y = 90°

(v) $$\angle$$y = 90° (Vertically opposite angles)
$$\angle$$x + $$\angle$$x + $$\angle$$y = 180° (Angle sum property of a triangle)
Or, 2 $$\angle$$x + 90° = 180°
Or, 2$$\angle$$x = 180° – 90°
Or, 2$$\angle$$x = 90°
$$\therefore$$ $$\angle$$x=90°/2=45°
Thus, $$\angle$$x = 45° and $$\angle$$y = 90

(vi) From the given figure, we have
$$\angle$$y = $$\angle$$x
$$\angle$$1 = $$\angle$$x
$$\angle$$2 = $$\angle$$x
Adding both sides, we have Adding both sides, we have Adding both sides, we have $$\angle$$y + $$\angle$$1 + $$\angle$$2 = 3$$\angle$$x
Or, 180° = 3$$\angle$$x [Angle sum property of a triangle]
$$\therefore$$ $$\angle$$x=180°/3=60°
$$\angle$$x = 60°, $$\angle$$y = 60°