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

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

【learning target】:

1 Proof of the sum of interior angles theorem of a triangle.

2 Master the sum of interior angles theorem of a triangle and initially learn to use auxiliary lines to prove problems

[Essential and difficult points to learn]

1 Key points: Proof of the sum of interior angles theorem of a triangle.

2 Difficulty: Proof method of triangle interior angle sum theorem.

General steps to prove a proposition:

(1) Understand the meaning of the question: distinguish the conditions of the proposition (known) and the 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.

Review and Thoughts

We know that the sum of the three interior angles of a triangle is equal to 180°. Do you still remember the exploration process of this conclusion?

(1) As shown in the figure, we moved ∠A to the position of ∠1 and ∠B to the position of ∠2. If we do not actually move ∠A and ∠B, do you have other ways to achieve the same effect?

(2) Based on the previous axioms and theorems, can you explain the proof of this conclusion in your own language? Can you write down the proof process in a simpler language? Communicate with your peers.

Appreciation of examples

Known: Figure 6-9, △ABC.

Prove: ∠A+∠B+∠C=180°

Analysis: Extend BC to D and draw ray CE∥AB through point C. This is equivalent to moving ∠A to the position of ∠1 and moving ∠B to the position of ∠2.

Prove: Draw the extension line CD of BC, and draw CE∥AB through point C, then

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

∠2= ∠B (the two straight lines are parallel and have equal angles).

And ∵∠1+∠2+∠3=1800 (the definition of a square angle),

∴ ∠A+∠B+∠ACB=1800 (equivalent substitution).

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Understand and understand mathematics from the perspective of motion changes

In △ABC, if BC does not move and "presses" point A toward BC, then as point A gets closer and closer to BC, ∠A becomes larger and larger (gets closer to 1800), while ∠B and ∠ C, getting smaller and smaller (getting closer to 00). What can you think of from this?

practice

1. As shown in the figure, it is known that AD is the common edge of △ABD and △ACD.

Prove: ∠BDC=∠BAC+∠B+∠C

Proof 1:

∵In △ABD, ∠1＝180°-∠B-∠3,

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

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

∴∠ BDC =360°-(180°-∠B-∠3)-(180°-∠C-∠4)

= ∠B+∠C+∠3+∠4.

And ∵ ∠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.

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