1. **Stating the problem:** We have a triangle with points A, L, C, T, B, D where segments AL, LC, and CT are equal in length. The angle $\angle TCD = 96^\circ$ is given, and we need to find the value of $x^\circ$, which is the angle at vertex A between segments AL and AB. 2. **Understanding the problem:** Since $AL = LC = CT$, points A, L, C, and T form a chain of equal segments. This suggests that triangle ALC and triangle LCT have some symmetry or equal sides. 3. **Key insight:** Because $AL = LC = CT$, triangle LCT is isosceles with $LC = CT$. The angle at C between CT and CD is given as $96^\circ$. 4. **Using the fact that the sum of angles around point C is 360°:** - The angle $\angle TCD = 96^\circ$ is given. - The angle $\angle LCT$ is between LC and CT, which are equal sides, so $\angle LCT$ is the vertex angle of isosceles triangle LCT. 5. **Calculate the base angles of triangle LCT:** Since $LC = CT$, the base angles $\angle CLT$ and $\angle CTL$ are equal. 6. **Sum of angles in triangle LCT:** $$\angle LCT + 2 \times \text{base angle} = 180^\circ$$ 7. **Relate $\angle LCT$ to $\angle TCD$:** Since $\angle TCD = 96^\circ$ and $\angle LCT$ is adjacent to it, the straight line at C implies: $$\angle LCT + 96^\circ = 180^\circ$$ Therefore, $$\angle LCT = 180^\circ - 96^\circ = 84^\circ$$ 8. **Calculate base angles:** $$2 \times \text{base angle} = 180^\circ - 84^\circ = 96^\circ$$ $$\text{base angle} = \frac{96^\circ}{2} = 48^\circ$$ 9. **Since AL = LC, triangle ALC is also isosceles with base AC:** The base angles at A and C in triangle ALC are equal. 10. **Calculate angle at A in triangle ALC:** The angle at C in triangle ALC is the base angle we found, $48^\circ$. Sum of angles in triangle ALC: $$\angle A + \angle L + \angle C = 180^\circ$$ Since $AL = LC$, $\angle A = \angle L$. 11. **Calculate $\angle A$:** $$2 \times \angle A + 48^\circ = 180^\circ$$ $$2 \times \angle A = 180^\circ - 48^\circ = 132^\circ$$ $$\angle A = 66^\circ$$ 12. **Therefore, the value of $x^\circ$ is:** $$x = 66^\circ$$