"Evidence by Contradiction" PPT courseware

"Evidence by Contradiction" PPT courseware

Short story: Bitter plum on the roadside

Once upon a time there was a smart boy named Wang Rong. When he was 7 years old, he went out to play with his friends and saw that the plum tree on the roadside was full of fruits. The friends went to pick the fruits one after another, but only Wang Rong stood still. Someone asked Wang Rong why,

Wang Rong replied: "If a tree grows by the road and has many children, the plum trees will suffer."

My friend picked one and tasted it, and it turned out to be a bitter plum.

How did Wang Rong know that plums are bitter?

What reasoning method did he use?

Prove: There cannot be two obtuse angles in a triangle.

Known: ΔABC.

Prove: There cannot be two obtuse angles in a triangle.

Proof: Suppose ΔABC has two obtuse angles,

Let's assume that ∠A and ∠B are both obtuse angles.

∵ ∠A+ ∠B �180 °

∴ ∠A+ ∠B+ ∠C �180 °

This contradicts "the sum of the interior angles of a triangle is 180°",

So, our assumption that there can be two obtuse angles in a triangle is wrong, therefore there cannot be two obtuse angles in a triangle.

Explore the steps again

1. Assume that the conclusion of the proposition is not valid

The conclusion that negates the original proposition must be rigorous to prevent improper negation or omissions.

2. Reasoning and argumentation, arriving at contradictions

The reasoning process must be complete, otherwise the truth or falsity of the proposition cannot be explained

3. The conclusion of the original proposition is established

Ability to find theorems, definitions or known conditions that create conflicts

Apply what you learn:

1. Use proof by contradiction to prove that "among the three interior angles of a triangle, at least one interior angle is less than or equal to 60°."

Proof: Suppose the three interior angles of the triangle are all greater than 60 degrees,

That is, ∠A�60°, ∠B�60°, ∠C�60°,

Then ∠ A+∠B+ ∠C�180° ,

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

∴It is not true that all three interior angles of a triangle are greater than 60°.

∴Among the three interior angles of a triangle, at least one interior angle is less than or equal to 60°

2. As shown in the figure, it is known that AB⊥EF is in M ​​and CD⊥EF is in N. Use proof by contradiction to prove: AB∥CD.

Class summary

What knowledge did you learn in this lesson?

1. What kind of proof method is called proof by contradiction?

2. What are the general steps to prove a proposition by proof by contradiction?

State the negative side of each of the following conclusions:

(1), a∥b

(2), a≥b

(3), b is a positive number

(4), a⊥b

(5), there is at least one

(6), there is at most one

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