1. **Problem statement:** Given that points A, E, and D are collinear and BCDE is a parallelogram, with \(\angle B = 100^\circ\) and \(\angle D = 51^\circ\), find the measure of \(\angle A\). 2. **Key properties:** - Opposite angles in a parallelogram are equal. - Adjacent angles in a parallelogram are supplementary (sum to \(180^\circ\)). - Since A, E, and D are collinear, \(\angle A\) is formed on the line through these points. 3. **Step 1: Identify angles in parallelogram BCDE.** - \(\angle B = 100^\circ\) (given). - Opposite angle \(\angle D = 100^\circ\) if BCDE is a parallelogram, but given \(\angle D = 51^\circ\), so this must be an interior angle at vertex D different from the parallelogram angle. 4. **Step 2: Use supplementary angles at vertex D.** - Since BCDE is a parallelogram, adjacent angles are supplementary. - \(\angle C + \angle D = 180^\circ\). - Given \(\angle D = 51^\circ\), so \(\angle C = 180^\circ - 51^\circ = 129^\circ\). 5. **Step 3: Use opposite angles equality.** - \(\angle B = \angle D\) in a parallelogram, but given values differ, so the given \(51^\circ\) at D is likely an angle formed by the line A-E-D and side of the parallelogram. 6. **Step 4: Analyze triangle formed by points A, E, and B.** - Since A, E, and D are collinear, \(\angle A\) is adjacent to \(\angle E\) and \(\angle D\). - The angle at A is supplementary to the sum of \(\angle B = 100^\circ\) and \(\angle D = 51^\circ\) because of the straight line through A, E, D. 7. **Step 5: Calculate \(\angle A\).** - \(\angle A = 180^\circ - (100^\circ - 51^\circ) = 180^\circ - 49^\circ = 131^\circ\) (incorrect, re-check). 8. **Step 6: Correct approach using parallelogram properties and collinearity.** - Since BCDE is a parallelogram, \(\angle B = \angle D = 100^\circ\) (opposite angles equal). - Given \(\angle D = 51^\circ\) must be an angle adjacent to the parallelogram, not the interior angle. - The angle at A is supplementary to the angle at E formed by the line A-E-D and side of the parallelogram. 9. **Step 7: Use the fact that \(\angle A + \angle B = 180^\circ\) (adjacent angles in parallelogram).** - \(\angle A = 180^\circ - 100^\circ = 80^\circ\). 10. **Step 8: Use triangle AED where \(\angle D = 51^\circ\) and \(\angle E = 100^\circ\) (since \(\angle B = 100^\circ\) and B and E are corresponding points). - Sum of angles in triangle AED is \(180^\circ\). - \(\angle A = 180^\circ - 100^\circ - 51^\circ = 29^\circ\). **Final answer:** \(\boxed{29^\circ}\) which corresponds to option B.