1(3). Write Can an obtuse and an acute angle make a straight angle? Can an obtuse and 11. Write Explain why testing a conjecture is not a generally accepted method of proof.

### 1. Can an obtuse and an acute angle make a straight angle? An obtuse angle is defined as an angle that measures more than 90 degrees but less than 180 degrees. An acute angle, on the other hand, is an angle that measures less than 90 degrees. A straight angle measures exactly 180 degrees. To determine if an obtuse angle and an acute angle can combine to form a straight angle, we can consider the following: - Let the obtuse angle be represented as \( x \) (where \( 90 < x < 180 \)). - Let the acute angle be represented as \( y \) (where \( 0 < y < 90 \)). When we add these two angles together, we have: \[ x + y \] For \( x + y \) to equal a straight angle, it must equal 180 degrees: \[ x + y = 180 \] Since \( x \) is greater than 90 degrees and \( y \) is less than 90 degrees, the sum \( x + y \) will always be greater than 90 degrees and less than 270 degrees. Therefore, it is possible for \( x + y \) to equal 180 degrees if the acute angle \( y \) is appropriately chosen. In conclusion, yes, an obtuse angle and an acute angle can combine to make a straight angle, provided that the acute angle is the right amount to complement the obtuse angle to reach 180 degrees. --- ### 11. Explain why testing a conjecture is not a generally accepted method of proof. Testing a conjecture involves checking specific cases or examples to see if the conjecture holds true. While this method can provide evidence that a conjecture may be true, it is not a generally accepted method of proof for several reasons: 1. **Limited Scope**: Testing a conjecture only verifies its truth for the specific cases that have been tested. There may be many other cases or scenarios where the conjecture does not hold true. A single counterexample is enough to disprove a conjecture. 2. **Inductive Reasoning**: Testing relies on inductive reasoning, which can lead to false conclusions. Just because a conjecture works for several examples does not guarantee it will work for all cases. Mathematical proofs require deductive reasoning, which provides certainty. 3. **Lack of Generality**: A proof must demonstrate that a statement is true in all cases, not just in a few. Testing does not provide a general argument that applies to all possible instances of the conjecture. 4. **Mathematical Rigor**: Mathematics values rigor and certainty. A proof provides a logical framework that establishes the truth of a statement based on axioms and previously established theorems, while testing does not offer the same level of assurance. In summary, while testing a conjecture can be a useful step in the exploration of mathematical ideas, it cannot replace the need for a formal proof that establishes the truth of a conjecture in a comprehensive and rigorous manner.