Answer:Direct proof, indirect proof, and proof by contrapositive
Explanation:
There are three different types of proofs commonly used in geometry: 1. Direct Proof: This type of proof involves providing a logical sequence of steps to demonstrate that a statement is true. In a direct proof, you start with the given information or known facts, and then use deductive reasoning to show how these facts lead to the desired conclusion. For example, to prove that the opposite sides of a parallelogram are congruent, you can use a direct proof by showing that the opposite sides are parallel and that they have the same length. 2. Indirect Proof: Also known as proof by contradiction, an indirect proof is used when it is difficult to prove a statement directly. Instead, you assume the opposite of what you want to prove and then show that this assumption leads to a contradiction or an impossible situation. By doing so, you can conclude that the opposite of the assumption must be true. For instance, to prove that the square root of 2 is irrational, you can use an indirect proof by assuming that it is rational and then showing that this leads to a contradiction. 3. Proof by Contrapositive: This type of proof is based on the contrapositive statement of a given statement. The contrapositive of a statement "If p, then q" is "If not q, then not p". Instead of proving the original statement directly, you prove its contrapositive, which is often easier to prove. If the contrapositive is true, then the original statement is also true. For example, to prove that if two angles are congruent, then their corresponding sides are proportional, you can use the contrapositive statement that if the corresponding sides are not proportional, then the angles are not congruent. These three types of proofs provide different approaches to proving statements in geometry. By understanding and applying them appropriately, you can strengthen your ability to reason logically and validate geometric claims.