"Why They Are Parallel" Proof PPT Courseware

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"Why They Are Parallel" Proof PPT Courseware

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This set of textbooks uses the following propositions as axioms:

1. If two straight lines are intercepted by a third straight line, if the angles are equal, then the two straight lines are parallel;

2. Two parallel lines are intercepted by a third straight line, and their angles are equal;

3. Two triangles whose sides and angles are equal are congruent;

4. Two triangles whose two angles and their included sides are equal are congruent;

5. Two triangles with three equal sides are congruent;

6. The corresponding sides of congruent triangles are equal and the corresponding angles are equal.

Axiom: If two straight lines are intercepted by a third straight line, if the parallel angles are equal, then the two straight lines are parallel. If the parallel angles are equal, the two straight lines are parallel.

Theorem: 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. If the interior angles on the same side are complementary, the two straight lines are parallel.

Known: As shown in the figure, ∠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,

∴∠1+∠2=180°.

∴∠1= 180° -∠2 ,

And ∵∠3+∠2=180°,

∴∠3= 180°-∠2.

∴∠1=∠3 ,

∴ a∥b.

Axioms, definitions, and proven theorems can all be used as a basis to prove new theorems.

practise:

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′.

Determine the shapes of these three quadrilaterals and explain your reasons.

Solution: ∵∠A+∠D= ∠α +∠β =180° (known)

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

The same principle can be proved: AD∥BC,

∴ ABCD is a parallelogram. (Definition of parallelogram)

That is, the three quadrilaterals sought are parallelograms.

2. Prove: Vertex angles are equal.

It is known that: straight lines AB and CD intersect at point O, ∠1 and ∠2 are opposite vertex angles,

Prove: ∠1=∠2.

Prove: ∵ ∠1+∠AOD=180 °, (1 straight angle = 180 °)

∠2+∠AOD=180°, (1 straight angle=180°)

∴ ∠1=∠2. (Supplementary angles of the same angle are equal)

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"Why They Are Parallel" Proof PPT Courseware 4:

"Why Are They Parallel" Proof PPT Courseware 4 Learning Objectives: (1) Preliminarily understand the basic steps and writing format of the proof (2) Prove that the internal angles on the same side are complementary according to the equal angles and two straight lines are parallel, and the internal angles of the two parallel lines are equal, Two straight lines are parallel and simple...

"Why They Are Parallel" Proof PPT Courseware 3:

"Why They Are Parallel" Proof PPT Courseware 3 Learning Objectives: 1. Preliminarily understand the basic steps and writing format of the proof. (Key points) 2. It will be proved that the internal angles on the same side are complementary based on the fact that the angles of the same angle are equal and the two straight lines are parallel. The two straight lines are parallel, the internal angles are equal and the two straight lines are parallel..

"Why They Are Parallel" Proof PPT Courseware 2:

"Why Are They Parallel" Proof PPT Courseware 2 We have explored the conditions for straight lines to be parallel before. 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] Two straight lines...

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Update Time: 2024-12-02

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