"Evidence by Contradiction" PPT Courseware 2

Description

"Evidence by Contradiction" PPT Courseware 2

learning target

1. Master the steps of proof by contradiction.

2. Be able to reason by proof by contradiction.

3. Learn the method of negative reasoning and cultivate the ability to reason from both positive and negative aspects.

Learning points

Proof steps by contradiction

Learning difficulties

Able to use proof by contradiction to make inferences and proofs

"There is at most one right angle in a triangle" Can you prove it?

Known: ΔABC

Prove: In ΔABC, if it contains a right angle, then it has only one right angle.

Proof: Suppose there are two (or three) right angles in ΔABC, let ∠A=∠B=90º

∵∠A+∠B=90º

∴∠A+∠B+∠C>180º

This contradicts "the sum of the interior angles of a triangle is 180º".

Therefore, the assumption that a triangle has two (or three) right angles is not true.

So, if a triangle contains right angles, it can only have one right angle.

The general steps to prove that a proposition is true by contradiction are:

The first step is to assume that the conclusion of the proposition does not hold.

The second step is to start from this assumption and other known conditions, and through reasoning and argumentation, arrive at results that are contradictory to the learned concepts, basic facts, proven theorems, properties, or problem conditions.

The third step is to determine if there are contradictory results that the hypothesis is not established, thereby indicating that the conclusion of the proposition is correct.

Commonly used expressions that negate each other:

is——existence——

Parallel - Vertical -

It’s equal to—— It’s all——

bigger than smaller than--

at least one--

There are at least three——

There are at least n——

At most one——

At most one of the triangles is a right angle——

give it a try

Prove by contradiction (fill in the blank): Among the interior angles of a triangle, at least one angle is greater than or equal to 60°.

Known: ∠A, ∠B, ∠C are the interior angles of △ABC.

Prove: At least one angle among ∠A, ∠B and ∠C is greater than or equal to 60°.

Proof: Assume that the conclusion being proved does not hold, that is

∠A ___ 60°, ∠B ___ 60°, ∠C ___60°

Then ∠A+∠B+∠C < 180°.

This contradicts the fact that the sum of the three interior angles of a triangle is equal to 180°.

Therefore, the hypothesis does not hold, and the conclusion verified holds.

Consolidation exercises

Prove the following proposition by proof by contradiction:

1. Two straight lines perpendicular to the same straight line are parallel

2. When two straight lines intersect, there is only one intersection point.

3. If two straight lines are parallel to a third straight line, then the two straight lines are also parallel to each other.

Keywords: Method of proof by contradiction teaching courseware, Hebei Education Edition eighth grade mathematics volume PPT courseware download, eighth grade mathematics slide courseware download, method of proof by contradiction PPT courseware download, .PPT format;

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Update Time: 2024-07-01

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