1. **Problem Statement:** Calculate the missing angles a, b, c, d, e, f, g, h, i, j, k, l in various configurations involving parallel lines and transversals. 2. **Key Concepts:** - Corresponding angles are equal. - Alternate interior angles are equal. - Co-interior angles sum to 180°. - Angles on a straight line sum to 180°. - Angles in a triangle sum to 180°. --- ### Top-left (a): 3. Given angle = 51°. 4. Since a is corresponding or alternate interior angle to 51°, $a = 51°$. --- ### Top-right (b, c): 5. Given angles: 104°, 117°, b°, c°. 6. Since 104° and b° are on a straight line, $b = 180° - 104° = 76°$. 7. Since 117° and c° are co-interior angles, $c = 180° - 117° = 63°$. --- ### Center-left (d, e): 8. Given angles: 67°, d°, e°, 75°. 9. d and 67° are alternate interior angles, so $d = 67°$. 10. e and 75° are alternate interior angles, so $e = 75°$. --- ### Center-right (f, g): 11. Isosceles triangle between parallel lines. 12. Exterior angle f equals sum of opposite interior angles. 13. Given f°, then $g = 180° - f$ (since g is exterior angle adjacent to f). --- ### Lower-left (h, i): 14. Isosceles triangle with base exterior angle 124°. 15. Interior opposite angles h and i are equal. 16. Sum of angles in triangle: $h + i + 124° = 180°$. 17. Since $h = i$, $2h + 124° = 180°$. 18. Solve: $2h = 56° \Rightarrow h = 28°$, so $i = 28°$. --- ### Bottom-left (j, k, l): 19. Triangle with apex angle 41°, base angles j, k, l, and exterior angle 119°. 20. Exterior angle 119° equals sum of two opposite interior angles. 21. Let base angles be j and k, then $j + k = 119°$. 22. Sum of triangle angles: $41° + j + k = 180°$. 23. Substitute $j + k = 119°$, so $41° + 119° = 160°$ which contradicts 180°. 24. Reconsider: l is exterior angle adjacent to k, so $l = 119°$. 25. Sum of triangle angles: $41° + j + k = 180°$. 26. Since $l = 119°$ is exterior to angle k, $k + 119° = 180° \Rightarrow k = 61°$. 27. Then $j = 180° - 41° - 61° = 78°$. --- **Final answers:** $$a=51°, b=76°, c=63°, d=67°, e=75°, f=\text{given}, g=180° - f, h=28°, i=28°, j=78°, k=61°, l=119°$$