"Why They Are Parallel" Proof PPT Courseware 4

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"Why They Are Parallel" Proof PPT Courseware 4

learning target:

(1) Have a preliminary understanding of the basic steps and writing format of proofs

(2) Be able to prove "the internal angles on the same side are complementary and the two straight lines are parallel" and "the interior angles on the same side are equal and the two straight lines are parallel" based on "the internal angles on the same side are equal and the two straight lines are parallel" and simply apply these conclusions.

(3) Feel the rigor of geometric reasoning and the determination of conclusions, and develop preliminary deductive reasoning abilities.

The first link: Preparation before class

Activities:

1.∠1 and ∠2 are ___________ angles.

2.______ and ______ are internally offset angles.

3._______________ is the interior angle on the same side.

4. In the same plane, two straight lines that do not intersect are called _______________.

5. If two straight lines are parallel to a third straight line, then the two straight lines _________ each other.

6.Two straight lines perpendicular to the same straight line____________.

7. If the angles of the same angle are equal, the two straight lines ________.

8. If the internal angles are equal, the two straight lines ____________.

9. If the interior angles on the same side are complementary, two straight lines _____________.

10. Two angles that add up to 180° are _______ each other, and they are also two angles that are complementary. That is, one angle is the supplement of another angle. For example: ∠4 and ______

The second link: Cooperation and exchange, exploring new knowledge

Proposition:

If two straight lines are intercepted by a third straight line, if the interior angles on the same side are complementary, then the two straight lines are parallel.

How to prove this proposition? (Work in groups to research, and explore ideas for proof based on the topic)

It is known that ∠1 and ∠2 are same-side interior angles intercepted by straight lines a and b by straight line c, and ∠1 and ∠2 are complementary. Prove: a∥b.

Analysis: To prove that straight lines a and b are parallel, you can think of applying the axiom of determination of parallel lines to prove it.

At this time, we can know from the figure: ∠1 and ∠3 are the same angles, so we only need to prove that ∠1=∠3, then a and b are parallel.

In this way, we have proved that a proposition is a true proposition through the process of reasoning. We call this true proposition: the determination theorem of parallel lines.

This theorem can be simply written as: interior angles on the same side are complementary and two straight lines are parallel.

Note: (1) The given axioms, definitions and proven theorems can be used as basis in the future. Used to prove new theorem.

(2) Every step of reasoning in the proof must be based on evidence, and cannot be "taken for granted". These grounds can be known conditions, definitions, axioms, or theorems that have been learned.

It is known that as shown in Figure 4, ∠1 and ∠2 are the internal offset angles of straight lines a and b intercepted by straight line c, and ∠1=∠2.

Prove: a∥b

Proof: ∵∠1=∠2 (known), ∠1+∠3=180° (right angle definition)

∴∠2+∠3=180° (equivalent substitution)

∴∠2 and ∠3 are complementary (definition of complementarity)

∴a∥b (interior angles on the same side are complementary and two straight lines are parallel).

In this way, we get another theorem 2 for determining that straight lines are parallel: if the interior angles are equal, the two straight lines are parallel.

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Update Time: 2024-09-24

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