Article: A mathematical proof is a series of logical statements supported by theorems and definitions that prove the truth of another mathematical statement. Proofs are the only way to know that a statement is mathematically valid. Being able to write a mathematical proof indicates a fundamental understanding of the problem itself and all of the concepts used in the problem. Proofs also force you to look at mathematics in a new and exciting way. Just by trying to prove something you gain knowledge and understanding even if your proof ultimately doesn't work. Before writing a proof, you need to think about the audience that you're writing for and what information they already know. If you are writing a proof for publication, you will write it differently than writing a proof for your high school math class. Knowing your audience allows you to write the proof in a way that they will understand given the amount of background knowledge that they have. There are a few different types of proofs and the one you choose depends on your audience and the assignment. If you're unsure which version to use, ask your teacher for guidance. In high school, you may be expected to write your proof in a specific format such as a formal two-column proof. A two-column proof is a setup that puts givens and statements in one column and the supporting evidence next to it in a second column. They are very commonly used in geometry. An informal paragraph proof uses grammatically correct statements and fewer symbols. At higher levels, you should always use an informal proof. The two-column proof is an easy way to organize your thoughts and think through the problem. Draw a line down the middle of the page and write all givens and statements on the left side. Write the corresponding definitions/theorems on the right side, next to the givens they support. For example: Angle A and angle B form a linear pair. Given. Angle ABC is straight. Definition of a straight angle. Angle ABC measures 180. Definition of a line. Angle A + Angle B = Angle ABC. Angle addition postulate. Angle A + Angle B = 180. Substitution. Angle A supplementary to Angle B. Definition of supplementary angles. Q.E.D. Using the two-column proof as a foundation, write the informal paragraph form of your proof without too many symbols and abbreviations. For example: Let angle A and angle B be linear pairs. By hypothesis, angle A and angle B are supplementary. Angle A and angle B form a straight line because they are linear pairs. A straight line is defined as having an angle measure of 180. Given the angle addition postulate, angles A and B sum together to form line ABC. Through substitution, angles A and B sum together to 180, therefore they are supplementary angles. Q.E.D.
Question: What is a summary of what this article is about?
Define mathematical proofs. Know your audience. Identify the type of proof you are writing. Write the two-column proof as an outline. Convert the two-column proof to an informal written proof.
Article: A lot of people act in ways that show disinterest and dislike without even realizing it. Figure out if there's anything that you door don't dothat may make you seem closed off, and try doing the opposite to look more open. Are you shy and quiet? Unfortunately, shyness is often misinterpreted by others as arrogance. Try to turn this assumption around by talking to people more often. What's your body language like? Many people look closed off without meaning to be. When you cross your arms, fidget, slouch, avoid eye contact, or rarely smile, you look less approachable to other people. Sometimes people aren't open due to negative past experiences, chemical imbalances, or neurological disorders. While self-help may work for some issues, others require the help of professionals to resolve. Depending on your problem, you may talk to a counselor, a therapist, your primary care physician, a psychiatrist, or some combination thereof. If you're having trouble opening up due to a fear of social situations, you may have social anxiety. Certain mental illnesses, neurodevelopmental disorders, and neuromuscular diseases can affect your ability to communicate nonverbally, making looking more open difficult. Some people may have trouble opening up because of past trauma or a lack of ability to trust others. A professional counselor can help you work through these issues and regain your ability to trust, allowing you to be more open. You don't have to change the positive or neutral aspects of your personality just to be more open. If you're comfortable with yourself and have fulfilling interpersonal relationships, then you might already be an open person in your own way. There's nothing wrong with being who you are. If you're introverted, don't try to be extroverted. Since many cultures are biased towards extroverts, it may be tempting to force yourself into a more extroverted persona. However, this can lead to burnout and unhappiness over time. Instead, find the right balance of social interaction that makes you happy and stick with it. If you're autistic, you're likely to have some trouble with typical nonverbal expression. You may want to learn how to do things like maintain appropriate eye contact and smile more often to help with social situations at school and work. Keep in mind that most people have to "perform" the same way on some level, even if it may come to them a little more naturally. Also, if you struggle with these things, don't beat yourself up over it. Just try your best.
Question: What is a summary of what this article is about?
Think about why you may be considered closed off. Consider talking to a professional. Remember that there are many types of open people.