"Why They Are Parallel" Proof PPT Courseware 3

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"Why They Are Parallel" Proof PPT Courseware 3

learning target:

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

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

3. Understand and master the general steps for proving propositions. (difficulty)

4. Feel the rigor of reasoning and the determination of conclusions in geometry, and develop preliminary deductive reasoning abilities.

Axiom If two straight lines are intercepted by a third straight line, if their co-position angles are equal, then the two straight lines are parallel.

Simply put: the same angles are equal and the two straight lines are 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.

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 (known)

∴∠1+ ∠2=180o (complementary definition)

∴∠1=180o- ∠2 (property of equation)

∵∠3+ ∠2=180o (1 square angle=180o)

∴∠3=180o- ∠2 (property of equation)

∴ ∠ 1 = ∠3 (equivalent substitution)

∴ a∥b (equal angles, two straight lines are 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.

Simply put: interior angles on the same side are complementary and two straight lines are parallel.

General steps to prove a proposition:

(1) Clarify the topic and conclusion;

(2) Draw corresponding graphics according to the meaning of the question;

(3) Write down what is known based on the question and conclusion and seek verification;

(4) Analyze the proof ideas and write down the proof process.

Exercise As shown in the figure: Lines AB and CD both intersect AE, and ∠1+∠A=180º.

Verification: AB//CD

Proof: ∵∠1+∠3=180 º (1 straight angle = 180 º)

∠2+∠3=180º (1 square angle=180º)

∴∠1=∠2 (equivalent substitution)

∵∠1+∠A=180º (known)

∴∠2+∠A=180º (equivalent substitution)

∴AB//CD

Kind tips:

1. You can use the axioms and theorems you have learned to prove;

2. Pay attention to the standardization of the proof format.

Extension and expansion:

1. As shown in the figure, it is a schematic diagram of a racing track. Among them, AB∥DE, measured ∠B=140°, ∠D=120°, then the degree of ∠C is ( )

A, 120° B, 100° C, 140° D, 90°

2. Known: As shown in the figure, E and F are two points on the diagonal AC of the quadrilateral ABCE, AF=CF, DF=BE, DF∥BE.

Verify: (1) △AFD≌△CEB.

(2) Quadrilateral ABCD is a parallelogram.

3. As shown in the figure, in the quadrilateral ABCD, the diagonals AC and BD intersect at point O, and OA/OC=OB/OD,

Prove: AB∥CD.

Class summary

General steps to prove a proposition:

(1) Clarify the title and conclusion;

(2) Draw corresponding graphics according to the meaning of the question;

(3) Write down what is known based on the question and conclusion and seek verification;

(4) Analyze the proof ideas and write down the proof process.

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Update Time: 2024-10-20

This template belongs to Mathematics courseware Beijing Normal University eighth grade mathematics volume 2 industry PPT template

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