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Authoritative PPT Summary
"What is a geometric proof" PPT courseware 2
teaching objectives
1. Understand the concepts of original proposition and converse proposition, be able to identify two mutually inverse propositions, and know that the original proposition is true but the converse proposition may not be true.
2. Prove the parallel lines determination theorem.
3. Cultivate students’ reasoning and argumentation abilities.
Please decide whether the following proposition is true or false
The shortest line segment between two points
Two straight lines are intercepted by a third straight line and have equal angles.
Preview test
1. Among two propositions, if the condition of the first proposition is the conclusion of the second proposition, and the conclusion of the first proposition is the condition of the second proposition, then these two propositions are called (reciprocal propositions)
2. The converse proposition of "the internal deviation angles are equal and the two straight lines are parallel" is (the two straight lines are parallel and the internal deviation angles are equal).
3. The converse of "the opposite vertex angles are equal" is (equal angles are the opposite vertex angles).
Is this converse proposition true or false? Give reasons.
think about it
What are the methods for determining parallel lines? do you remember
1. The parallel angles are equal and the two straight lines are parallel.
2. The internal angles are equal and the two straight lines are parallel.
3. Interior angles on the same side are complementary and two straight lines are parallel.
One certificate
The internal angles are equal and the two straight lines are parallel.
Known: As shown in the figure, ∠1 and ∠2 are the internal offset angles obtained by intercepting straight lines a and b by straight line c, ∠1=∠2.
Prove: a∥b
Proof: ∵∠2=∠3 (the vertex angles are equal)
∠1=∠2 (known)
∴∠1=∠3 (equivalent substitution)
∴a∥b (the angles are equal and the two straight lines are parallel)
Interior angles on the same side are complementary and two straight lines are parallel.
Known: As shown in the figure, ∠1 and ∠2 are the same side interior angles obtained by intercepting straight lines a and b by straight line c, ∠1+∠2=180°.
Prove: a∥b
Prove: ∵∠2+∠3=180 (definition of supplementary angle)
∠1+∠2=180° (known)
∴∠1=∠3 (supplementary angles of the same angle are equal)
∴ a∥b (the angles are equal and the two straight lines are parallel)
change
1. The internal angles are equal and the two straight lines are parallel.
2. Interior angles on the same side are complementary and two straight lines are parallel.
What is the converse of the above two propositions?
1. Two straight lines are parallel and their internal offset angles are equal.
2. Two straight lines are parallel and the interior angles on the same side are complementary.
Two propositions in which conditions and conclusions are interchanged are called reciprocal propositions. One of the propositions is called the original proposition, and the other is called the converse proposition of the original proposition.
collaborative inquiry
If the original proposition is a true proposition, must its converse proposition be a true proposition?
Precautions:
1. A proposition must have a converse proposition.
2. The converse of a proposition is not necessarily a true proposition.
3. If the converse of a theorem is also a true proposition, then the converse is the converse of the original theorem.
Pointing examples:
As shown in the figure, △ABC is a roof truss, AB=AC, AD is the bracket connecting point A and midpoint D of BC.
Verify: △ABD≌△ACD.
Proof: ∵ Point D is the midpoint of BC (known)
∴BD=CD (the meaning of the midpoint of the line segment)
And ∵AB=AC (known)
AD=AD (common edge)
∴△ABD≌△ACD (SSS)
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"What is a Geometric Proof" PPT courseware:
"What is Geometric Proof" PPT courseware learning objectives 1. Understand the meaning of proof and know the meaning of theorem. 2. Preliminarily understand the three steps of geometric proof, understand the writing format of geometric proof through examples, and feel that every step of reasoning in the proof process must be well-founded.