"Why They Are Parallel" Proof PPT Courseware 2

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"Why They Are Parallel" Proof PPT Courseware 2

Earlier we explored the conditions for straight lines to be parallel.

Let’s think about it: under what circumstances are two straight lines parallel to each other?

[1] In the same plane, two straight lines that do not intersect are called parallel lines.

[2] If two straight lines are parallel to a third straight line, then the two straight lines are parallel to each other.

[3] If the same angles are equal, the two straight lines are parallel. If the interior angles are equal, the two straight lines are parallel. If the interior angles on the same side are complementary, the two straight lines will be parallel.

Theorem: 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.

Interior angles on the same side are complementary and two straight lines are parallel.

→As shown in the figure, ∠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.

So how to prove this question? Let's analyze it.

To prove that straight lines a and b are parallel, you can think of applying the parallel line determination axiom to prove. At this time, we can know from the figure: ∠1 and ∠3 are co-position angles, so we only need to prove that ∠1=∠3, then a and b That is parallel.

Because it can be seen from the figure that ∠2 and ∠3 form a right angle, that is, ∠2+∠3=180°, so: ∠3=180°-∠2. And because ∠2 and ∠1 are complementary among the known conditions, that is : ∠2+∠1=180°, so ∠1=180°-∠2, so by equivalent substitution we can know: ∠1=∠3.

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

Proof: ∵∠1 and ∠2 are complementary (known)

∴∠1+∠2=180° (definition of complementarity)

∴∠1=180°-∠2 (property of equation)

∵∠3+∠2=180° (flat angle=180°)

∴∠3=180°-∠2 (property of equation)

∴∠1=∠3 (equivalent substitution)

∴a∥b (the angles are equal and the two straight lines are parallel)

Reflection: Interior angles on the same side are complementary, and two straight lines are parallel. → It is known that ∠1 and ∠2 are interior angles on the same side when straight lines a and b are intercepted by straight line c, and ∠1 and ∠2 are complementary. Prove: a∥b.

Reflection: 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 to prove new theorems.

(2) Every step of reasoning in the proof must have a basis and cannot be "taken for granted". These basis can be known conditions, definitions, axioms, or theorems that have been learned. When learning the proof for the first time, it is required to write down the basis. in parentheses after each step of reasoning.

Practice in class

1. The bottom of the hive is surrounded by three congruent quadrilaterals. The shape of each quadrilateral is as shown in the figure, where ∠α=109°28′ and ∠β=70°32′. Try to determine the shapes of these three quadrilaterals. , and explain your reasons.

Solution: The shape of these three quadrilaterals is a parallelogram.

The reason is: ∵∠α=109°28′∠β=70°32′ (known)

∴∠α+∠β=180° (property of equation)

∴AB∥CD, AD∥BC (interior angles on the same side are complementary and two straight lines are parallel)

∴ Quadrilateral ABCD is a parallelogram (definition of parallelogram)

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Update Time: 2024-09-08

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