I'm stuck on these 2 problems. My Geometry teacher won't help me.
What is the inverse of the Isosceles Triangle Theorem (ITT)?
What is the contrapositive of the Isosceles Triangle Theorem (ITT)?
Here is how I remember this:
1. hypothesis
If a cow, then a mammal.
assume true
2. converse
If a mammal, then a cow.
maybe
3. inverse
If not a cow, then not a mammal.
maybe
4. contrapositive
If not a mammal, then not a cow.
true
You can also draw Venn diagram to remember this.
So
I am not sure how your book stated theorem, there are several "isosceles triangle theorems.
here is perhaps the most likely:
Hypothesis:
If two sides equal, then opposite angles equal.
Inverse
If two sides not equal, then opposite angles not equal
Contrapositive
If two angles not equal, then opposite sides not equal.
hmm.. okay, but i'm not really seeing how that goes with the ITT
Tell me what ITT is exactly.
Put your ITT in the form
IF THIS
THEN THAT
That is the hypothesis. The rest just follows the recipe.
Okay, the ITT is
If a triangle has two congruent sides, then the angles oppositeof those sides are congruent
Okay. I figured the problem out now. Thanks for your help!
Good, I figured you would !
To find the inverse and contrapositive of a theorem, we need to understand what these terms mean in logic.
1. Inverse of the Isosceles Triangle Theorem (ITT):
The inverse of a theorem is formed by negating both the hypothesis and the conclusion of the original theorem.
The Isosceles Triangle Theorem states that "If a triangle has two equal sides, then it has two equal angles."
To find the inverse of this theorem, we need to negate both the hypothesis and the conclusion. The negation of "a triangle has two equal sides" is "a triangle does not have two equal sides." Similarly, the negation of "it has two equal angles" is "it does not have two equal angles."
Therefore, the inverse of the Isosceles Triangle Theorem (ITT) is:
"If a triangle does not have two equal sides, then it does not have two equal angles."
2. Contrapositive of the Isosceles Triangle Theorem (ITT):
The contrapositive of a theorem is formed by swapping the hypothesis and the conclusion of the inverse statement.
In the inverse statement we found above, the hypothesis is "a triangle does not have two equal sides" and the conclusion is "it does not have two equal angles."
To find the contrapositive, we need to swap the hypothesis and the conclusion. So, the contrapositive of the Isosceles Triangle Theorem (ITT) is:
"If a triangle does not have two equal angles, then it does not have two equal sides."
Remember, the inverse and contrapositive of a theorem may or may not hold true. They are just different statements derived from the original theorem by negating and swapping the hypothesis and conclusion.