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"Proposition and Proof" PPT Courseware 2
Reflection:
Which propositions are in the following sentences? Which ones are not propositions?
And decide whether the following propositions are true or false.
(1) The supplementary angles of the same angle are equal.
(2) Congruent angles are opposite vertex angles.
(3) Pick any point C on the straight line AB.
(4) The sum of two sides of a triangle is greater than the third side.
(5) Two triangles with equal areas are congruent.
(6) If a>b, then ac>bc.
If two straight lines are intercepted by a third straight line, then the two straight lines are parallel if the angles of parallelism are equal.
If two straight lines are intercepted by a third straight line, if the two straight lines are parallel, then the parallel angles are equal.
Like this, two propositions in which the condition and conclusion of one proposition are the conditions and conclusion of another proposition are called reciprocal propositions.
Among two mutually inverse propositions, if we call one of the propositions the original proposition, then the other proposition is the converse proposition of the original proposition.
Please write the converse of the following proposition, and indicate whether the original proposition and the converse are true or false.
1. Two straight lines are intercepted by a third straight line. If the internal offset angles are equal, then the two straight lines are parallel.
2. If two angles are opposite vertex angles, then the two angles are equal.
3. If a number is divisible by 3, then the number is also divisible by 6.
4. Two numbers a and b are known. If a+b>0, then a-b>0.
Propositions include true propositions and false propositions. To show that a proposition is false, just give a counterexample.
To prove that a proposition is true, one must start from the conditions of the proposition and conduct well-founded reasoning based on the basic facts, definitions, properties, and theorems that have been learned. This process of reasoning is called proof.
Example: Prove that two straight lines parallel to the same straight line are parallel.
Known: As shown in the figure, straight lines a, b, c, a∥c, b∥c
Prove: a∥b.
Proof: As shown in the figure, draw a straight line d that intersects the straight lines a, b, and c respectively.
∵a∥c(known)
∴∠1=∠3 (two straight lines are parallel and have equal angles)
∵b∥c
∴∠2=∠3 (Two straight lines are parallel and have equal angles.)
∴∠1=∠3 (equivalent substitution)
∴a∥b (the angles are equal and the two straight lines are parallel)
That is, two straight lines parallel to the same straight line are parallel.
If the converse of a theorem is a true proposition, then this converse can also be called the converse of the original theorem.
A theorem and its converse are reciprocal theorems.
For example, "Two straight lines are parallel and their internal offset angles are equal."
"The internal offset angles are equal and the two straight lines are parallel."
Give yourself an example
Vertical angles are equal
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"Proposition and Proof" PPT courseware:
"Proposition and Proof" PPT courseware, group discussion and independent exploration. Exchange ( ) and ( ) of a proposition to form a new proposition. If the original proposition is called the original proposition, then this new proposition is called the converse proposition of the original proposition. These two propositions are called...