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"Definition and Proposition" Proof PPT Courseware 2
Definition
It can be seen that communication must have a common understanding of certain names and terms. To this end, it is necessary to describe the meaning of the names and terms and make clear regulations, that is, to give their definitions.
For example:
1. "Persons with the nationality of the People's Republic of China are called citizens of the People's Republic of China", which is the definition of ________.
2. "The length of the line segment between two points is called the distance between the two points" is the definition of ________.
3. ____________________ is the definition of "parallelogram".
think
Compare the expression forms of the following sentences. Which one makes a judgment about the matter? Which ones don’t make judgments about things?
(1) Birds are animals.
(2) If a2=b2, then a=b.
(3) 0.33 is an irrational number.
(4) The two straight lines are parallel and have equal angles.
Observing the following propositions, can you find what characteristics these propositions have in common?
1. If the three sides of two triangles are equal, then the two triangles are congruent.
2. If two angles are opposite vertex angles, then the two angles are equal.
3. If a triangle is an isosceles triangle, then the two base angles of the triangle are equal.
4. If two parallel lines are intercepted by a third straight line, then the parallel angles are equal.
structure of proposition
Example: Find the conditions and conclusions of the proposition, and rewrite them in the form of "if..., then...":
1. Opposite angles are equal.
Condition: the two angles are opposite vertex angles,
Conclusion: These two angles are equal.
Rewrite: If two angles are opposite vertex angles, then the two angles are equal.
2. Two triangles with three equal sides are congruent.
Condition: three sides are equal,
Conclusion: These two triangles are congruent.
Rewrite: If two triangles have three corresponding equal sides, then the two triangles are congruent.
Which of these propositions are correct? What is incorrect? How do you know they are incorrect?
1. If two angles are equal, then they are opposite angles;
2. If a>b, b>c, then a=c;
3. Two triangles whose two angles and the opposite side of one of the angles are equal are congruent;
4. If the outdoor temperature is below 0℃, the water on the ground will definitely freeze.
5. The areas of congruent triangles are equal.
Axioms of this textbook
1. Two points determine a straight line.
2. The shortest line segment between two points.
3. In the same plane, there is and is only one straight line perpendicular to the known straight line through a point.
4. Two straight lines are intercepted by a third straight line. If the angles are equal, then the two straight lines are parallel.
5. There is only one straight line parallel to this straight line through a point outside the straight line.
6. Two triangles whose sides and angles are equal are congruent.
7. Two triangles whose two angles and included sides are equal are congruent.
8. Two triangles with three equal sides are congruent.
Other axioms
The relevant properties of equations and the relevant properties of inequalities can be regarded as axioms
In an equation or inequality, a quantity can be replaced by its equivalent. For example, if, then, this property is also regarded as an axiom and is called "equivalent substitution".
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"Definition and Proposition" Proof PPT Courseware 5:
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