The worksheet you're working on involves logic statements related to geometry. Specifically, it deals with conditional statements and their variations: converse, inverse, and contrapositive. Let's break down the content:

Conditional Statement:

• Given Statement (P → Q): "If an angle is obtuse, then it measures 137°."
  • P: "An angle is obtuse."
  • Q: "The angle measures 137°."

Converse Statement (Q → P):

• Q → P: "If it measures 137°, then the angle is obtuse."
  • The truth of this statement depends on whether the angle measuring 137° always makes it obtuse, which is true because obtuse angles are angles greater than 90° and less than 180°, and 137° fits in this range. Therefore, the converse statement is True.

Inverse Statement (¬P → ¬Q):

• ¬P → ¬Q: "If an angle is not obtuse, then it does not measure 137°."
  • This statement implies that only obtuse angles can measure 137°, which is correct, because non-obtuse angles are either acute or right, and neither of them can measure 137°. Thus, the inverse statement is True.

Contrapositive Statement (¬Q → ¬P):

• ¬Q → ¬P: "If it does not measure 137°, then the angle is not obtuse."
  • This statement is logically equivalent to the original conditional statement. If the angle doesn’t measure 137°, it does not necessarily mean that the angle isn’t obtuse (e.g., an angle measuring 120° is still obtuse). So this statement would be False.

Truth Values:

• Conditional Statement (P → Q): False (Not all obtuse angles measure exactly 137°).
• Converse Statement (Q → P): True (An angle measuring 137° is obtuse).
• Inverse Statement (¬P → ¬Q): True (An angle not obtuse cannot be 137°).
• Contrapositive Statement (¬Q → ¬P): False (Not all obtuse angles must measure exactly 137°).

Let me know if you need further clarification or have additional questions! Here are some related questions to deepen your understanding:

1. What is a biconditional statement, and can it apply to this example?
2. Can an acute angle ever have a converse, inverse, or contrapositive like the ones listed here?
3. How would the truth values change if the original conditional statement were about a right angle instead?
4. How can you generalize the contrapositive of a conditional statement?
5. How does the contrapositive compare to the converse in logical structure?

Tip: Remember, the contrapositive of a statement is always logically equivalent to the original statement!