"Proof of the Sum of the Interior Angle Theorem of a Triangle" Proof PPT Courseware 3

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"Proof of the Sum of the Interior Angle Theorem of a Triangle" Proof PPT Courseware 3

cognitive reasoning

The so-called inductive reasoning is reasoning that draws general conclusions from individual knowledge. Inductive reasoning: Based on the fact that some objects of a class of things have a certain property, inference that all objects of this class of things have this property is called inductive reasoning (referred to as induction). Induction is a process from the specific to the general. It belongs to reasonable reasoning. Inductive reasoning is good at discovering conclusions.

For example: in a plane, the sum of the interior angles of a right triangle is 180 degrees; the sum of the interior angles of an acute triangle is 180 degrees; the sum of the interior angles of an obtuse triangle is 180 degrees; right triangles, acute triangles and obtuse triangles are all triangles; therefore, everything in the plane The sum of the interior angles of a triangle is 180 degrees.

Deductive reasoning is reasoning that derives specific conclusions from general premises. Deductive reasoning has forms such as syllogism, hypothetical reasoning, and disjunctive reasoning.

Review and Thoughts

General steps to prove a proposition:

(1) Understand the meaning of the question: analyze the proposition (known) and conclusion (verification);

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

(3) Combined with graphics, use symbolic language to write "known" and "verified";

(4) Analyze the meaning of the question and explore proof ideas;

(5) Based on the idea, use mathematical symbols and mathematical language to clearly write out the proof process;

(6) Check whether the expression process is correct and complete.

Proof method one

Triangle interior angle sum theorem: The sum of the three interior angles of a triangle is 180°

Known: Figure △ABC.

Prove: ∠A+∠B+∠C=180°.

Prove: PQ∥BC through point A, then

∠1=∠B (the two straight lines are parallel and the internal offset angles are equal),

∠2=∠C (the two straight lines are parallel and the internal offset angles are equal),

And ∵∠1+∠2+∠3=180° (the definition of a straight angle),

∴ ∠BAC+∠B+∠C=180° (equivalent substitution).

The sum of the interior angles of a triangle theorem The sum of the three interior angles of a triangle is equal to 180°.

In △ABC, ∠A+∠B+∠C=180°.

Several deformations of ∠A+∠B+∠C=180°:

∠A=180° –(∠B+∠C).

∠B=180° –(∠A+∠C).

∠C=180° –(∠A+∠B).

∠A+∠B=180°-∠C.

∠B+∠C=180°-∠A.

∠A+∠C=180°-∠B.

practice

1. As shown in the figure, it is known that AD is the common edge of △ABD and △ACD. Verify: ∠BDC=∠BAC+∠B+∠C

Proof 1:

∵In △ABD, ∠1＝180°-∠B-∠3,

In △ADC, ∠2＝180°-∠C-∠4 (triangle interior angle sum theorem),

And ∵∠BDC=360°-∠1-∠2 (definition of circumferential angle)

∴∠BDC =360°-(180°-∠B-∠3)-(180°-∠C-∠4)

= ∠B+∠C+∠3+∠4.

Also ∵ ∠BAC＝∠3+∠4,

∴ ∠BDC＝∠B+∠C+∠BAC (equivalent substitution)

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For more information about the PPT courseware "Proof of the sum of the interior angles of a triangle theorem", please click on the Proof ppt tag of the proof of the sum of the interior angles of a triangle theorem.

"Proof of the Sum of the Interior Angle Theorem of a Triangle" Proof PPT Courseware 4:

"Proof of the sum of the interior angles of a triangle theorem" Proof PPT courseware 4 [Learning objectives]: 1. Proof of the sum of the interior angles of a triangle theorem. 2. Master the theorem of the sum of the interior angles of a triangle, and initially learn to use auxiliary lines to prove problems [important and difficult points in learning] 1. focus: triangles The sum of interior angles theorem...

"Proof of the Sum of the Interior Angle Theorem of a Triangle" Proof PPT Courseware 2:

"Proof of the sum of the interior angles of a triangle theorem" Proof PPT courseware 2 Teaching objectives: Knowledge and skills: 1. Understand the sum of the interior angles of a triangle theorem; 2. Master the proof method of the sum of the interior angles of a triangle theorem; 3. Be able to use the sum of the interior angles of a triangle theorem to prove and solve Other related...

"Proof of the Sum of Interior Angles Theorem of a Triangle" Proof PPT Courseware:

"Proof of the Sum of the Interior Angles of a Triangle Theorem" Proof PPT courseware review and thinking about the general steps to prove a proposition: (1) Understand the meaning of the question: distinguish the conditions (known) and conclusion (verification) of the proposition; (2) Draw the graph according to the meaning of the question; (3) Use symbolic language to write what is known and verified based on graphics;...

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Update Time: 2024-10-20

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