Explaining the isosceles theorem.
Explain why the SAS Inequality theorem is also called the Hinge theorem.
the steps for Indirect Proofs and using an indirect proof with an algebraic problem.
(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?
Make sense to me.
Good job!
Thanks!
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.
Explaining the isosceles theorem:
The isosceles theorem states that if a triangle has two sides that are congruent (equal in length), then the angles opposite those two sides are also congruent. In other words, if you have a triangle with two sides of equal length, then the angles opposite those sides will also be equal.
To understand why this is true, you can imagine folding the triangle along the line that connects the two congruent sides. When you do this, the two congruent sides will overlap perfectly, forming a line segment. Since the two sides are congruent, the fold line will also bisect the angle between them, creating two congruent angles.
Therefore, whenever you have a triangle with two sides of equal length, you can automatically conclude that the angles opposite those sides are also equal. This is the essence of the isosceles theorem.
Explaining why the SAS Inequality theorem is also called the Hinge theorem:
The SAS (Side-Angle-Side) Inequality theorem states that if two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the third side of the first triangle is longer than the third side of the second triangle.
This theorem is also commonly referred to as the Hinge theorem because it involves a hinge-like movement. When you have two triangles that satisfy the conditions of the SAS Inequality theorem, you can think of one triangle as being "hinged" at the vertex of the included angle, and the other triangle rotates around that hinge.
As the second triangle rotates, the third side (which is not directly connected to the hinge) can only get closer or farther from the vertex of the included angle. If the angles and side lengths satisfy the conditions of the SAS Inequality theorem, the third side of one triangle will always be longer than the third side of the other triangle as it rotates around the hinge.
Because of this visual image of a hinge-like movement, the SAS Inequality theorem is also commonly known as the Hinge theorem.
Steps for indirect proofs and using an indirect proof with an algebraic problem:
Indirect proofs, also known as proof by contradiction, are a way of proving a statement by assuming that the statement is false and then deriving a contradiction or inconsistency from that assumption. Here are the steps for an indirect proof:
1. Start with the statement or proposition you want to prove.
2. Assume the opposite of the statement is true (i.e., assume the statement is false).
3. Derive some logical consequences from the assumption.
4. Look for a contradiction or inconsistency in the logical consequences.
5. If a contradiction is found, it means the assumption (that the statement is false) must be incorrect.
6. Therefore, the original statement must be true.
To illustrate how to use an indirect proof with an algebraic problem, let's consider an example:
Problem: Prove that there are infinitely many prime numbers.
Solution (using indirect proof):
1. Start with the statement: "There are infinitely many prime numbers."
2. Assume the opposite: "There are finitely many prime numbers."
3. If there are finitely many primes, let's say p₁, p₂, ..., pₙ are all the prime numbers.
4. Consider the number N = p₁ * p₂ * ... * pₙ + 1.
5. N is not divisible by any of the primes p₁, p₂, ..., pₙ because it leaves a remainder of 1 when divided by any of them.
6. Therefore, N must be either prime or divisible by a prime that is not in the list p₁, p₂, ..., pₙ.
7. In both cases, N contradicts our assumption that there are finitely many primes.
8. Thus, our assumption was incorrect, and there must be infinitely many prime numbers.
By following these steps, we have used an indirect proof to demonstrate that there are infinitely many prime numbers.