Posted by Rachel on Thursday, February 14, 2008 at 4:52pm.
(1) A triangle is called an isosceles triangle if it has two sides with equal lengths. Consider an isosceles triangle ABC with AC = CB. Then side AB (i.e. the side that is not equal to the other sides) is called the base side. The angle opposite the base side is called the vertex angle and the other two angles of an isosceles triangle are called base angles.
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(2) The SAS Inequality Theorem (Hinge Theorem) states:
If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.
(3) The SSS Inequality Theorem (Converse of Hinge Theorem) states:
If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.
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Steps in an Indirect Proof:
1-Assume that the opposite of what you are trying to prove is true.
2-From this assumption, see what conclusions can be drawn. These conclusions must be based upon the assumption and the use of valid statements.
3-Search for a conclusion that you know is false because it contradicts given or known information. Oftentimes you will be contradicting a piece of GIVEN information.
4-Since your assumption leads to a false conclusion, the assumption must be false.
5-If the assumption (which is the opposite of what you are trying to prove) is false, then you will know that what you are trying to prove must be true.
Now, use these steps and form your own algebraic indirect question.
Then write back and we will check over your work.
Done!
Given: 2x-3>7
Prove:x>5
Assume: x<5 or x=5
Using a table with several possibilities for x given that x<5 or x=5
x - 2x-3
1 = -1
2 = 1
3 = 4
4 = 5
5 = 7
It's a contradiction because then x<5 or when x>5, 2x-3< or = 7
So in both cases, the assumption leads to the contradiction of a known fact. Therefore, the assumption that x is < or = to 5 must be false, which means that x>5 must be true.
Is that right?
yes
Given: 2x-3 > 7
Prove: x > 5
Proof:
Let it be given that 2x - 3 > 7.
Assume that x < 5.
Then by the Addition Property of Equality:
2x - 3 > 7
+3 +3
2x > 10
By the Division Property of Equality;
2x /2 > 10/2
By Simplification: x > 5.
But this is a contradiction to the assumption that x < 5.
Thus x > 5.
QED.
This is a more formal proof. Using a table of values only shows the case for the values you selected. In reality, you must show the case is true for all values greater than 5. The paragraph proof above does just that.
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