1. **Find the value of x in each figure:** (i) Angles on a straight line sum to 180°. $$60° + x° + 60° = 180°$$ $$x = 180° - 120° = 60°$$ (ii) Angles on a straight line sum to 180°. $$3x° + 2x° + 120° = 180°$$ $$5x = 60° \Rightarrow x = 12°$$ (iii) Angles on a straight line sum to 180°. $$35° + x° + 60° = 180°$$ $$x = 180° - 95° = 85°$$ (iv) Angles around a point sum to 360°. $$83° + x° + 47° + 92° + 75° = 360°$$ $$x = 360° - 297° = 63°$$ (v) Angles around a point sum to 360°. $$3x° + 2x° + x° + 2x° = 360°$$ $$8x = 360° \Rightarrow x = 45°$$ (vi) Vertically opposite angles are equal. $$105° = 3x° \Rightarrow x = 35°$$ 2. **Line l || m and transversal n cuts them at P and Q. Given ∠1 = 75°, find all other angles:** Corresponding angles are equal, alternate interior angles are equal, and angles on a straight line sum to 180°. At P: - ∠1 = 75° (given) - ∠2 = 105° (linear pair: 180° - 75°) - ∠3 = 75° (vertically opposite to ∠1) - ∠4 = 105° (vertically opposite to ∠2) At Q: - ∠5 = 75° (corresponding to ∠1) - ∠6 = 105° (linear pair with ∠5) - ∠7 = 75° (vertically opposite to ∠5) - ∠8 = 105° (vertically opposite to ∠6) 3. **Lines l, m, n are parallel and cut by transversal p. Find ∠1, ∠2, ∠3:** Given angle near line l is 120° adjacent to ∠1. - ∠1 = 60° (linear pair: 180° - 120°) - ∠2 = 60° (corresponding to ∠1) - ∠3 = 60° (corresponding to ∠1) 4. **Find the value of x:** (i) Interior angles on the same side of transversal are supplementary. $$109° + x° = 180° \Rightarrow x = 71°$$ (ii) Interior angles on the same side of transversal are supplementary. $$35° + x° = 180° \Rightarrow x = 145°$$ (iii) Angles around a point sum to 360°. $$83° + 47° + 92° + 75° + x° = 360°$$ $$x = 360° - 297° = 63°$$ (iv) Parallel lines cut by transversal create equal alternate interior angles. Given 50° at top intersection, x opposite it is also 50°. 5. **Definitions:** (i) Complementary angles: Two angles whose sum is 90°. (ii) Supplementary angles: Two angles whose sum is 180°. (iii) Adjacent angles: Two angles that share a common side and vertex. (iv) Linear pair: A pair of adjacent angles whose non-common sides form a straight line. (v) Vertically opposite angles: Angles opposite each other when two lines intersect; they are equal. 6. **In Fig. 80, line AC || DE, ∠ABD = 32°, ∠E = 122°, find x and y:** Since AC || DE and BD is transversal: - ∠ABD = 32° (given) - ∠E = 122° (given) - ∠x and ∠y are angles at E and D respectively. Using supplementary angles: $$x = 180° - 122° = 58°$$ Since AC || DE, alternate interior angles are equal: $$y = 32°$$ 7. **In Fig. 83, PQ || RS, find x:** Given: - ∠B = 55° - ∠C = 130° Since PQ || RS and BD is transversal, angles at B and C are supplementary: $$55° + 130° = 185°$$ (which is inconsistent, so likely ∠C is exterior angle) Assuming triangle ABC: Sum of angles in triangle: $$x + 55° + 130° = 180°$$ $$x = 180° - 185° = -5°$$ (impossible) Re-examining, if ∠C = 130° is exterior angle, then interior angle at C is: $$180° - 130° = 50°$$ Sum of interior angles: $$x + 55° + 50° = 180°$$ $$x = 75°$$ **Final answers:** 1. (i) 60°, (ii) 12°, (iii) 85°, (iv) 63°, (v) 45°, (vi) 35° 2. ∠1=75°, ∠2=105°, ∠3=75°, ∠4=105°, ∠5=75°, ∠6=105°, ∠7=75°, ∠8=105° 3. ∠1=60°, ∠2=60°, ∠3=60° 4. (i) 71°, (ii) 145°, (iii) 63°, (iv) 50° 5. See definitions above. 6. x=58°, y=32° 7. x=75°