Final answer:
The most likely diagnosis for the patient with a rash and skin thickening resembling an orange-peel on her legs is Hashimoto's thyroiditis, as it is associated with symptoms of hypothyroidism, including myxedema.
Step-by-step explanation:
The 45-year-old female with a rash and thickening of the skin on her legs exhibiting an orange-peel appearance may be exhibiting signs of myxoedema, which is strongly associated with hypothyroidism. The most common cause of hypothyroidism in the United States is Hashimoto's thyroiditis, which is an autoimmune condition where the thyroid gland is attacked by the immune system. Given the provided options, Hashimoto's thyroiditis (a) is the most likely diagnosis that correlates with the described skin findings. Hashimoto's thyroiditis typically presents with multiple symptoms including thickened skin, a condition known as myxedema.