"If Two Straight Lines Are Parallel" Proof PPT Courseware

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"If Two Straight Lines Are Parallel" Proof PPT Courseware

1 According to "two parallel lines are intercepted by a third straight line, the internal offset angles are equal". Can you make related figures?

2. Can you write down what you know and verify based on the graph you made?

3 Can you explain the idea of ​​​​the proof?

It is known that the two straight lines are parallel and the same angles are equal. Prove that the internal deviation angles are equal.

Two parallel lines intersected by a third straight line have equal interior angles.

It is known that, as shown in Figure 6-23, straight lines a∥b, ∠1 and ∠2 are internal offset angles of straight lines a and b intercepted by straight line c.

Prove: ∠1=∠2.

Can you talk about the idea of ​​​​the proof?

Proof: ∵a∥b (known)

∴∠3=∠2 (two straight lines are parallel and have equal angles)

∵∠1=∠3 (the two vertex angles are equal)

∴∠1=∠2 (equivalent substitution)

Reflection: After proving that this proposition is true, we can call it a theorem. That is, the property theorem of parallel lines. In this way, we can use it as the basis for future proofs.

Two parallel lines are intercepted by a third straight line, and their interior angles are complementary.

It is known that, as shown in Figure 6-23, the straight lines a∥b, ∠1 and ∠2 are the interior angles on the same side of the straight lines a and b intercepted by the straight line c.

Prove: ∠1+∠2=180°

Proof: ∵a∥b (known)

∴∠3=∠2 (two straight lines are parallel and have equal angles)

∵∠1+∠3=180° (1 straight angle=180°)

∴∠1+∠2=180° (equivalent substitution)

Reflection: Through the process of reasoning, it is proved that the proposition "Two parallel lines are intercepted by a third straight line, and the interior angles on the same side are complementary" is a true proposition. We call it the theorem, that is, the property theorem of parallel lines, which can be directly applied in the future to prove other conclusions.

1. Prove that the bisectors of adjacent supplementary angles are perpendicular to each other.

Known: As shown in Figure 6-25, ∠AOB and ∠BOC are complementary angles to each other, OE bisects ∠AOB, and OF bisects ∠BOC.

Prove: OE⊥OF.

Proof: ∵OE bisects ∠AOB.

OF bisects ∠BOC (known)

∴∠EOB=∠AOB

∠BOF=∠BOC (definition of angle bisector)

∵∠AOB+∠BOC=180° (1 straight angle=180°)

∴∠EOB+∠BOF= (∠AOB+∠BOC)=90° (property of equation)

That is ∠EOF=90°

∴OE⊥OF (definition of vertical)

Lesson Summary

In this lesson, we mainly study the proof of the property theorem of parallel lines, and summarize the general steps of the proof.

1. Properties of parallel lines:

Axiom: Two straight lines are parallel and have equal angles.

Theorem: Two straight lines are parallel and their interior angles are equal.

Theorem: Two straight lines are parallel and interior angles on the same side are complementary.

General steps for proof:

(1) Draw the graph according to the meaning of the question.

(2) Based on the conditions and conclusions, combined with graphics, write what is known and verified.

(3) After analysis, find out the way to prove from what is known, and write down the proof process.

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Update Time: 2024-10-02

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