1. **Stating the problem:** Given that \(\angle 3 = 74^\circ\), find the measures of other related angles formed by two parallel lines \(l\) and \(m\) cut by two transversals. 2. **Important rules:** - Corresponding angles are equal. - Alternate interior angles are equal. - Consecutive interior angles are supplementary (sum to 180°). 3. **Using the given information:** - Since \(\angle 3 = 74^\circ\) and \(\angle 3\) and \(\angle 6\) are vertically opposite angles, \(\angle 6 = 74^\circ\). 4. **Finding \(\angle 1\):** - \(\angle 1\) and \(\angle 3\) are corresponding angles (since lines \(l\) and \(m\) are parallel), so \(\angle 1 = 74^\circ\). 5. **Finding \(\angle 8\):** - \(\angle 3\) and \(\angle 8\) are supplementary (they form a linear pair), so $$\angle 8 = 180^\circ - 74^\circ = 106^\circ.$$ 6. **Finding \(\angle 2\):** - \(\angle 2\) and \(\angle 6\) are vertically opposite angles, so \(\angle 2 = 74^\circ\). 7. **Finding \(\angle 4\):** - \(\angle 4\) and \(\angle 8\) are corresponding angles, so \(\angle 4 = 106^\circ\). 8. **Finding \(\angle 5\):** - \(\angle 5\) and \(\angle 1\) are corresponding angles, so \(\angle 5 = 74^\circ\). 9. **Finding \(\angle 7\):** - \(\angle 7\) and \(\angle 4\) are vertically opposite angles, so \(\angle 7 = 106^\circ\). **Final answers:** - \(\angle 1 = 74^\circ\) - \(\angle 2 = 74^\circ\) - \(\angle 3 = 74^\circ\) (given) - \(\angle 4 = 106^\circ\) - \(\angle 5 = 74^\circ\) - \(\angle 6 = 74^\circ\) - \(\angle 7 = 106^\circ\) - \(\angle 8 = 106^\circ\)