What you miss in using a free, grammar-translation based program is cultural context. "The madam" sounds weird in English, but "a senhora" does not in Portuguese. (See all the other comments on the thread about how "a senhora" is used.)
So we can giggle about how in English, "the madam" means the manager of a brothel and we'd never use that phrase outside of a certain context, but the correct translation of "a senhora" into English would just be "madam," or the far more common "ma'am." (For those who would contend that "madam" is only used to mean the manager of a brothel, I would point you to the example of Judge Judy, who, if you've ever watched her, calls women "madam" all the time and never uses it to imply that they work in a house of ill repute.)
I do not know whether Duo accepts "madam" or "ma'am" as translations for "a senhora," but they should. As for "we are a mirror," I have no explanation for that.
Im a little surprised at the comments. Madam is a good translation. It is very formal but still used in English (at least in the US) much like "a senhora" is used in Portuguese. Yes, it can mean the owner of a brothel, but also: It is used as a greeting in business/formal letters, such as, "Dear Sir or Madam,..." If/when we have a female president, she would be addressed as "Madam President." Also, formally, or even just very politely/respectfully, addressing a woman. The less formal "ma'am" (mam) may be more common but that is merely short for "madam."
We have already had Madam Secretaries Rice, Albright and Clinton.
The use of "madam" is, however, quite limited in the United States. It would be considered "hyper-formal" as a form of address in daily life.
"Ma'am" is used extensively in the southern US, along with "sir". "Yes, ma'am" / "yes, sir" - responding to figures of authority.
I totally agree with gregdois. As in the USA, "Madam" is used as a polite form of address in the UK and Ireland, especially in department stores and restaurants (e.g. when a shop assistant or waiter is addressing an adult female customer. It is also always used in a letter where the writer does not know who the recipient will be, as in, "Dear Sir/Madam" or "Dear Sir or Madam" as gregdois has explained. (By the way, we don't tend to use "Ma'am" in this part of the world and we have already had two "Madam Presidents" in Ireland :-) )
The third person honorific with the definite article (O senhor, a senhora, and plural counterparts) act as the second person pronouns in European Portuguese. "Voce" and "Voces" are used (if at all) for those with whom one is acquainted; "tu" is for intimates. A lady would be addressed properly as "A senhora dona [first name], as in "Lady Diana".
Naturally in English we have degraded the only two respectful forms of address for a non-noblewoman, "mistress" (from which "Miss", "Mrs." and ultimately "Ms." derive) and "madam", to mean something dirty.
This is one of the cases where cultural differences make it difficult to find an equivalent and equal translation. "A senhora" is commonly used in Brazil to address an older lady as a mark of respect. My Brazilian wife may address her mother as "a senhora" in certain circumstances, but I would never have addressed my mother as "Madam" in any circumstance.