"Definition and Proposition" PPT Courseware 2

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"Definition and Proposition" PPT Courseware 2

communication and discovery

We have explored many mathematical conclusions in the past, some positive and some negative. Can you give a few examples of each?

If two angles are opposite vertex angles, then the two angles are congruent.

If two angles are not equal, then they are not subtended angles

All of these are statements that make judgments about something. Statements that express judgments like this are called propositions.

Example 1: State the conditions and conclusions of the following propositions

(1) If the three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent;

Solution (1) Condition: The three sides of one triangle are equal to the three sides of another triangle.

Conclusion: These two triangles are congruent

(2) If the two sides and one angle of a triangle are equal to the two sides and one angle of another triangle, then the two triangles are congruent.

Solution: Condition: Two sides and one angle of a triangle are equal to two sides and one angle of another triangle.

Conclusion: These two triangles are congruent.

(3) Two straight lines are intercepted by a third straight line. If the angles are equal, then the two straight lines are parallel;

Condition: Two straight lines intersected by a third straight line have equal angles

Conclusion: Two straight lines are parallel

(4) The two base angles of an isosceles triangle are equal

First change this proposition into the form of "if...then..."

If two angles are the two base angles of an isosceles triangle, then the two angles are congruent.

Condition: The two angles are the two base angles of an isosceles triangle

Conclusion: These two angles are equal.

(1) If the three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent;

(2) If the two sides and one angle of a triangle are equal to the two sides and one angle of another triangle, then the two triangles are congruent

(3) Two straight lines are intercepted by a third straight line. If the angles are equal, then the two straight lines are parallel.

(4) The two base angles of an isosceles triangle are equal

When the conditions of a proposition are true, the conclusion must also be true. It is called a true proposition.

In other words, a correct proposition is a true proposition

Among the four propositions in Example 1, are there any propositions for which the conclusion is incorrect even if the conditions are met? If so, indicate which one it is?

(2) in Example 1: When the conditions of the proposition are true, there is no guarantee that the conclusion of the proposition will always be true.

In other words, an incorrect proposition is a false proposition.

Expansion and extension

Write the conditions and conclusions of the following propositions, and determine which ones are false. If they are false, please give a counterexample.

1. The supplementary angle of an angle is greater than the angle

2. If the product of two rational numbers is less than zero, then the sum of the two numbers is also less than zero.

3. Two straight lines perpendicular to the same straight line are perpendicular

4. The hypotenuse of a right triangle is larger than any other right-angled side.

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Update Time: 2024-11-22

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