the main > Borehole pumps > The sum of the angles of the triangle is the solution. “Solution of problems on the application of the theorem on the sum of the angles of a triangle and the theorem on the outer angle of a triangle. Building a drawing and a short statement of the theorem

The sum of the angles of the triangle is the solution. “Solution of problems on the application of the theorem on the sum of the angles of a triangle and the theorem on the outer angle of a triangle. Building a drawing and a short statement of the theorem

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Slide captions:

7th grade. Solving problems. "The sum of the angles of a triangle. The outer angle of a triangle"

8 9 10 11 12 14 15 16 17 18 20 21 22 23 24 1 2 3 4 5 6 13 19 7 ... according to ready-made drawings

The theorem on the sum of the angles of a triangle. A B C The sum of the angles of a triangle is 180 0.

The outer corner of the triangle. Property. A B C The outer angle of a triangle is equal to the sum of two angles of a triangle that are not adjacent to it. D

Properties isosceles triangle... А М В К С N Angles at the base. Median, height, bisector. In an isosceles triangle, the angles at the base are equal. In an isosceles track, the bisector drawn to the base is the median and the height.

Medians, bisectors and heights of triangles. A K B M C R O N L S H Median Bisector Height

B A O C Adjacent corners

Equilateral triangle. A B C In an equilateral triangle, all sides are EQUAL and all angles are EQUAL.

1. Answer Hint (3) Properties of an isosceles triangle Find the angles of an isosceles track if the angle at the base is 2 times the angle opposite the base. The sum of the angles of the triangle C A B x 2x 2x

2. Answer Hint (3) Outside corner of a triangle Find the angles of an isosceles track if the angle at the base is 3 times less than the outside angle adjacent to it. The sum of the angles of the triangle C A B x 3x The property of the outer angle of the triangle

3. Answer 50 0 C A B Given: ∆ ABC, AB = BC, AD - bisector, Find: Hint (4) Properties of isosceles triangle Bisector of triangle D? Sum of the angles of a triangle Adjacent angles

4. Answer 7 5 0 К С Given: ∆ CDE, DK is the bisector, Find the angles of the triangle CDE. Hint (3) Consider ∆ CDK Triangle bisector D Sum of triangle angles 28 0 E

five . Answer 50 0 M A Given: ∆ ABC, BM - height, Find the angle CBM. Hint (3) Properties of an isosceles triangle Height of an isosceles triangle B Sum of angles of triangle C

6. Answer 12 0 0 C A B Given: ∆ ABC, AB = BC = 5 cm, Find: AC Hint (4) Properties of an isosceles triangle Outer corner of a triangle Adjacent angles D Equilateral triangle

Solving problems based on ready-made drawings. It is necessary to write down the condition of the problem from the picture and answer the question. There are no prompts in tasks. 8 9 1 0 7 1 1 1 2 14 15 1 6 13 1 7 1 8 20 21 22 23 24 19

7. Answer 3 0 0 A Find: B C?

8. Answer 4 0 0 A Find: B C D? ? ?

nine . Answer 30 0 D A BC = AC Find: B C?

10. Answer 110 0 A Find: B C 40 0? ?

Lesson objectives:

to acquaint students with the theorem on the sum of the angles of a triangle, to classify triangles by angles;
consider the application of the theorem to solving problems.

Lesson Objectives:

Educational:

formulate and consider the plan of the proof of the theorem on the sum of the angles of a triangle;
to classify triangles by angles;
consider the problem of applying the proven statement.

Developing: the ability to analyze, generalize the knowledge gained, develop mathematical speech.

Educational:

foster cognitive activity, culture of communication;
foster respect for the historical heritage in the field of mathematics.

Lesson type: partially exploratory.

Method: research using theoretical knowledge.

Equipment:

multi-projector;
presentation;
handout, task - a card for working out the theorem when solving problems.

Interdisciplinary connections: history.

The use of health-saving technologies in the lesson:

change of activities;
development of auditory and visual analyzers in every child.

Lesson plan:

1. Organizational moment.

Hello, sit down. (Presentation. Slide 1)

Yes, the path of knowledge is not smooth
But we know from school years
There are more riddles than answers
And there is no limit to the search.

2. Updating knowledge.

Let's remember everything that is needed in today's lesson.

DBE - expanded.

Slide 2.

2) Properties of an isosceles triangle. Find 1.

1 = 70 °

Formulate the opposite of the isosceles triangle property.

3) properties of parallel lines.

Slide 4

2 = 43 ° 1 = 60 °

- Like criss-crossing corners.

4) Introductory task. Slide 5

ABF - isosceles

B = 30 °, AF BD,

BD - bisector CBF

sum of angles ABF

Is the sum of the angles ABF coincidentally equal to 180 °, or does any triangle possess this property? ( Any triangle has a sum of angles equal to 180 °.)

This statement is called the triangle sum theorem.

So, the topic of the lesson: The sum of the angles of a triangle. Slide 6, 7, 8.

Often a preschooler also knows
What is a triangle.
And how can you not know ...
But it's a completely different matter -
Very fast and skillful
The magnitudes of all angles
Find out in the triangle.

To find quickly and correctly the angles in any triangle, you need to consider the theorem on the sum of all the angles of a triangle. This is what we will do now in the lesson.

Objectives:

- consider the outline of the proof of the theorem on the sum of the angles of a triangle;
- to classify triangles by angles;
- learn to apply the theorem on the sum of the angles of a triangle when solving problems.

Historical background on the "sum of the angles of a triangle" theorem.

The property of the sum of the angles of a triangle was empirically, that is, it was established empirically, probably back in Ancient Egypt, but the information that has come down to us about its various proofs refers to a later time. The proof set forth in modern textbooks is contained in Proclus's commentary on Euclid's Beginnings. Slides 9,10.

The angles of a triangle add up to 180 °

Prove:

A + B + C = 180 °

Proof plan:

Because in the condition of the theorem there is not enough data for the proof, then the question arises of introducing an auxiliary element (an additional construction is the construction of a straight line). The same situations arise when there is not enough data to solve problems.

a) Construct DE AC through vertex B ABC
b) Mark 1, 2, 3.

2) Prove that A = 1, C = 3

A = 1 as criss-crossing angles at DE AC,

AB - secant.

3) Prove that 1 + 2 + 3 = 180 °;

hence, A + 2 + C = 180 °

DBE - expanded

So 1 + 2 + 3 = 180 °

And since as criss-crossing angles at DE AC

Hence, A + 2 + C = 180 °

The theorem is proved.

4) What triangles are distinguished on the sides? (Isosceles, equilateral, versatile.)

Triangles are classified not only by the sides, but also by the corners. Let's talk about the corners first.

- What is an angle? (An angle is a shape formed by two rays emanating from the same point. The rays are called the sides of the angle, and the point is the apex of the angle.)
- What is the right angle? (An angle of 90º.)
- What angle is called unfolded? (An angle of 180º.)
- What angle is called acute? (Angle less than 90º.)
- What angle is called obtuse? (Angle greater than 90 ° but less than 180 °.)

Thus, the corners are sharp, straight, obtuse, unfolded.

Draw three corners in your notebook: sharp, blunt, and straight. Complete the drawing to the triangle.

- What needs to be done for this? (Take a point on the sides of the corner and connect them.)
- What are the triangles? (Obtuse, rectangular, acute-angled.)

Slide 13–16.

Oral test: Slide 17 the test is taken - "Lesson development on geometry grade 7, Gavrilova NF, M .: VAKO, 2006".

1) In a triangle ABC, A = 90 °, while the other two angles can be:

a) one is sharp and the other can be straight;
b) both are sharp;
c) one is sharp and the other can be blunt.

2) In a triangle ABC, B is obtuse, while the other two angles can be:

a) only sharp;
b) sharp and straight;
c) sharp and dull.

3) An acute-angled triangle may contain:

a) all corners are sharp;
b) one obtuse and 2 acute angles;
c) one straight line and 2 acute angles.

Checking by Slide 18, 19, 20.

5) Cards with the task are issued. The time for self-fulfillment is assigned - 7 minutes. Then it is verified through multimedia.

Practicing skills using ready-made drawings: Slide 21-30.

Find 1, 2.

6)Lesson conclusion:

- By types of angles are considered (acute-angled, obtuse, right-angled triangle).

- What is the sum of the angles in any triangle (The sum of the angles in any triangle is 180 °).

- We will also consider this theorem when solving problem No. 228 (a)

Recorded: House. task: Ch. IV §1 p. 30 No. 223 (a; b), 228 (b).

No. 228 (a). Consider: 2 cases of solving the problem:

If there is time test.

The sum of the angles of a triangle

The umma of the angles of an arbitrary triangle is 180 degrees.