"Focus on the exterior angles of a triangle" Proof PPT courseware 3

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"Focus on the exterior angles of a triangle" Proof PPT courseware 3

Triangle interior angle sum theorem

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=1800:

∠A=180° –(∠B+∠C).

∠B=180°–(∠A+∠C).

∠C=180°–(∠A+∠B).

∠A+∠B=180°-∠C.

∠B+∠C=180°-∠A.

∠A+∠C=180°-∠B.

As shown in the figure. ∠1 is an exterior angle of △ABC. What is the relationship between ∠1 and other angles in the figure?

Can you prove your conclusion?

∠1+∠4=180°;

∠1>∠2;

∠1>∠3;

∠1=∠2+∠3.

Prove: ∵∠2+∠3+∠4=180° (triangle interior angle sum theorem),

∠1+∠4=180° (the meaning of a straight angle),

∴∠1= ∠2+∠3. (Equal substitution).

∴ ∠1>∠2,∠1>∠3 (the sum is greater than the part).

Expressed in words:

An exterior angle of a triangle is equal to the sum of its two non-adjacent interior angles.

An exterior angle of a triangle is greater than any interior angle that is not adjacent to it.

connotation and denotation

Here, we directly derive two new theorems through the triangle interior angle sum theorem. Like this, a theorem directly derived from an axiom or theorem is called a corollary of this axiom or theorem.

Corollaries can be used as theorems.

Corollary to the sum of angles theorem of a triangle:

Corollary 1: An exterior angle of a triangle is equal to the sum of its two non-adjacent interior angles.

Corollary 2: An exterior angle of a triangle is greater than any interior angle that is not adjacent to it.

Practical applications:

1. The shape of a part is as shown in the figure. According to regulations, ∠A should be equal to 90°, ∠B and ∠C should be 21° and 32° respectively. If the inspection worker measures ∠BDC=148°, the part is judged to be unqualified. .Use the knowledge you have learned about triangles to explain the reasons why the parts are unqualified.

2. On the green field, a football player attacks with the ball and always tries his best to rush towards goal AB. Please explain the reason why he does this based on the knowledge you have learned.

Practice in class

It is known that the regular five-pointed star on the national flag is as shown in the figure. Find the degree of: ∠A+∠B+∠C+∠D+∠E.

Analysis: Try to use the exterior angles to "gather" these five angles into a triangle, and use the triangle interior angle sum theorem to solve the problem.

Solution: ∵∠1 is an exterior angle of △BDF (the meaning of exterior angle),

∴ ∠1=∠B+∠D (an exterior angle of a triangle is equal to the sum of its two non-adjacent interior angles).

And ∵ ∠2 is an exterior angle of △EHC (the meaning of exterior angle),

∴ ∠2=∠C+∠E (an exterior angle of a triangle is equal to the sum of its two non-adjacent interior angles).

And ∵∠A+∠1+∠2=180° (triangle interior angle sum theorem).

∴ ∠A+∠B+∠C+∠D+∠E =180° (equality property).

Summary and expansion

Understand geometric propositions

Proof methods, steps, formats and precautions.

Triangle interior angle sum theorem

The sum of the three interior angles of a triangle is equal to 180°. In △ABC, ∠A+∠B+∠C=180°.

Corollary 1: An exterior angle of a triangle is equal to the sum of its two non-adjacent interior angles.

Corollary 2: An exterior angle of a triangle is greater than any interior angle that is not adjacent to it.

Focus on the exterior angles of the triangle.

Corollary 3: The two acute angles of a right triangle are complementary to each other.

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Update Time: 2024-09-11

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