I have a 50 question assigment that its due tomorrow and I'm stuck on this question.
"Write the contrapositive,converse,and inverse of this statement.Determine whether they are true or false.If its false.find a conterexample
"Two angles measure 90 are compmentary".
Please help
To understand the question and answer it correctly, we need to know the definitions of contrapositive, converse, and inverse of a statement.
1. Contrapositive: The contrapositive of a statement is formed by negating both the hypothesis and conclusion of the original statement and flipping them. If the original statement is of the form "If p, then q," then the contrapositive is "If not q, then not p."
2. Converse: The converse of a statement is formed by switching the hypothesis and the conclusion of the original statement. If the original statement is of the form "If p, then q," then the converse is "If q, then p."
3. Inverse: The inverse of a statement is formed by negating both the hypothesis and the conclusion of the original statement. If the original statement is of the form "If p, then q," then the inverse is "If not p, then not q."
Now, let's work on the given statement "Two angles measure 90 are complementary":
1. Original Statement: "Two angles measure 90 are complementary."
- A hypothesis: "Two angles measure 90"
- A conclusion: "are complementary"
2. Contrapositive:
- Negating the hypothesis: "Two angles do not measure 90"
- Negating the conclusion: "are not complementary"
- Contrapositive Statement: "If two angles do not measure 90, then they are not complementary."
3. Converse:
- Switching the hypothesis and conclusion: "If two angles are complementary, then they measure 90."
Note: This is only true if all complementary angles measure 90 degrees. However, not all angles that measure 90 degrees are necessarily complementary, so the converse is not always true.
4. Inverse:
- Negating the hypothesis: "Two angles do not measure 90"
- Negating the conclusion: "are not complementary"
- Inverse Statement: "If two angles do not measure 90, then they are not complementary."
Now, let's evaluate the truth value of each statement:
- Original Statement: "Two angles measure 90 are complementary."
- This statement is true, as two angles measuring 90 degrees are indeed complementary.
- Contrapositive Statement: "If two angles do not measure 90, then they are not complementary."
- This statement is also true, as angles that do not measure 90 degrees are not complementary.
- Converse Statement: "If two angles are complementary, then they measure 90."
- This statement is not always true. There could be angles that are complementary and don't measure 90 degrees.
- Inverse Statement: "If two angles do not measure 90, then they are not complementary."
- This statement is true, as angles that do not measure 90 degrees are indeed not complementary.
So, the original statement and its contrapositive are true, while the converse is not always true.