Theorem on the sum of the interior angles of a triangle

The sum of the angles of a triangle is 180°.

Proof:

• Triangle ABC is given.
• Draw a line DK through the vertex B parallel to the base AC.
• \angle CBK= \angle C as internal crosswise lying with parallel DK and AC, and secant BC.
• \angle DBA = \angle A internal crosswise lying at DK \parallel AC and secant AB. Angle DBK is straight and equal to
• \angle DBK = \angle DBA + \angle B + \angle CBK
• Since the straight angle is 180 ^\circ , and \angle CBK = \angle C and \angle DBA = \angle A , we get 180 ^\circ = \angle A + \angle B + \angle C.

Theorem proven

Consequences from the theorem on the sum of angles of a triangle:

1. The sum of the acute angles of a right triangle is 90°.
2. In an isosceles right triangle, each acute angle is 45°.
3. In an equilateral triangle, each angle is 60°.
4. In any triangle, either all angles are acute, or two angles are acute, and the third is obtuse or right.
5. An exterior angle of a triangle is equal to the sum of two interior angles that are not adjacent to it.

Triangle exterior angle theorem

An exterior angle of a triangle is equal to the sum of the two remaining angles of the triangle that are not adjacent to that exterior angle.

Proof:

• Triangle ABC is given, where BCD is the exterior angle.
• \angle BAC + \angle ABC +\angle BCA = 180^0
• From the equalities, the angle \angle BCD + \angle BCA = 180^0
• We get \angle BCD = \angle BAC+\angle ABC.

>>Geometry: The sum of the angles of a triangle. Complete Lessons

TOPIC OF THE LESSON: The sum of the angles of a triangle.

Lesson Objectives:

• Consolidation and testing of students' knowledge on the topic: "The sum of the angles of a triangle";
• Proof of the properties of the angles of a triangle;
• The use of this property in solving the simplest problems;
• The use of historical material for the development of cognitive activity of students;
• Instilling the skill of accuracy in the construction of drawings.

Lesson objectives:

• Check students' ability to solve problems.

Lesson plan:

1. Triangle;
2. Theorem on the sum of the angles of a triangle;
3. Task example.

Triangle.

File:O.gif Triangle- the simplest polygon having 3 vertices (corners) and 3 sides; a part of a plane bounded by three points and three line segments connecting these points in pairs.
Three points in space that do not lie on one straight line correspond to one and only one plane.
Any polygon can be divided into triangles - this process is called triangulation.
There is a section of mathematics entirely devoted to the study of the patterns of triangles - Trigonometry.

Theorem on the sum of the angles of a triangle.

File:T.gif The triangle sum of angles theorem is a classical theorem in Euclidean geometry that states that the sum of the angles of a triangle is 180°.

Proof" :

Let Δ ABC be given. Let us draw a line parallel to (AC) through the vertex B and mark the point D on it so that the points A and D lie on opposite sides of the line BC. Then the angle (DBC) and the angle (ACB) are equal as internal crosses lying at parallel lines BD and AC and the secant (BC). Then the sum of the angles of the triangle at vertices B and C is equal to the angle (ABD). But the angle (ABD) and the angle (BAC) at vertex A of triangle ABC are interior one-sided with parallel lines BD and AC and secant (AB), and their sum is 180°. Therefore, the sum of the angles of a triangle is 180°. The theorem has been proven.

Consequences.

The exterior angle of a triangle is equal to the sum of the two angles of the triangle that are not adjacent to it.

Proof:

Let Δ ABC be given. The point D lies on the line AC so that A lies between C and D. Then BAD is external to the angle of the triangle at the vertex A and A + BAD = 180°. But A + B + C = 180°, and hence B + C = 180° – A. Hence BAD = B + C. The corollary is proved.

Consequences.

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

Task.

The external angle of a triangle is the angle adjacent to any angle of this triangle. Prove that an exterior angle of a triangle is equal to the sum of two angles of the triangle that are not adjacent to it.
(Fig.1)

Decision:

Let in Δ ABC ∠DAC be external (Fig.1). Then ∠DAC=180°-∠BAC (according to the property of adjacent angles), according to the theorem on the sum of angles of a triangle ∠B+∠C =180°-∠BAC. From these equalities we get ∠DAC=∠B+∠C

Interesting fact:

The sum of the angles of a triangle :

In Lobachevsky's geometry, the sum of the angles of a triangle is always less than 180. In Euclid's geometry, it is always equal to 180. In Riemannian geometry, the sum of the angles of a triangle is always greater than 180.

From the history of mathematics:

Euclid (III century BC) in the work “Beginnings” gives the following definition: “Parallel are straight lines that are in the same plane and, being extended indefinitely in both directions, do not meet with each other on either side” .
Posidonius (1st century BC) "Two straight lines lying in the same plane, equidistant from each other"
The ancient Greek scientist Pappus (III century BC) introduced the symbol of parallel lines - sign =. Subsequently, the English economist Ricardo (1720-1823) used this symbol as an equal sign.
Only in the 18th century did they begin to use the symbol of parallel lines - the sign ||.
The live connection between generations is not interrupted for a moment, every day we learn the experience accumulated by our ancestors. The ancient Greeks, based on observations and practical experience, drew conclusions, expressed hypotheses, and then, at meetings of scientists - symposiums (literally "feast") - they tried to substantiate and prove these hypotheses. At that time, the statement was formed: "Truth is born in a dispute."

Questions:

1. What is a triangle?
2. What does the triangle sum theorem say?
3. What is the outer angle of the triangle?

Theorem. The sum of the interior angles of a triangle is equal to two right angles.

Take some triangle ABC (Fig. 208). Let us denote its interior angles by 1, 2 and 3. Let us prove that

∠1 + ∠2 + ∠3 = 180°.

Let us draw through some vertex of the triangle, for example B, the line MN parallel to AC.

At vertex B, we got three angles: ∠4, ∠2 and ∠5. Their sum is a straight angle, therefore, it is equal to 180 °:

∠4 + ∠2 + ∠5 = 180°.

But ∠4 \u003d ∠1 are internal cross-lying angles with parallel lines MN and AC and a secant AB.

∠5 = ∠3 are internal cross lying angles with parallel lines MN and AC and secant BC.

Hence, ∠4 and ∠5 can be replaced by their equals ∠1 and ∠3.

Therefore, ∠1 + ∠2 + ∠3 = 180°. The theorem has been proven.

2. Property of the external angle of a triangle.

Theorem. An exterior angle of a triangle is equal to the sum of two interior angles that are not adjacent to it.

Indeed, in triangle ABC (Fig. 209) ∠1 + ∠2 = 180° - ∠3, but also ∠BCD, the external angle of this triangle, not adjacent to ∠1 and ∠2, is also equal to 180° - ∠3 .

Thus:

∠1 + ∠2 = 180° - ∠3;

∠BCD = 180° - ∠3.

Therefore, ∠1 + ∠2= ∠BCD.

The derived property of the external angle of a triangle refines the content of the previously proved theorem on the external angle of a triangle, in which it was stated only that the external angle of a triangle is greater than each internal angle of the triangle that is not adjacent to it; now it is established that the external angle is equal to the sum of both internal angles not adjacent to it.

3. Property of a right triangle with an angle of 30°.

Theorem. The leg of a right triangle opposite an angle of 30° is equal to half the hypotenuse.

Let the angle B be equal to 30° in a right-angled triangle ACB (Fig. 210). Then its other acute angle will be 60°.

Let us prove that the leg AC is equal to half of the hypotenuse AB. We continue the leg AC beyond the vertex of the right angle C and set aside the segment CM, equal to the segment AC. We connect point M with point B. The resulting triangle BCM is equal to triangle DIA. We see that each angle of the triangle AVM is equal to 60°, therefore, this triangle is equilateral.

The AC leg is equal to half of AM, and since AM is equal to AB, the AC leg will be equal to half of the hypotenuse AB.

. (Slide 1)

Lesson type: lesson learning new material.

Lesson Objectives:

• Educational:
  • consider the sum of triangle angles theorem,
  • show the application of the theorem in solving problems.
• Educational:
  • fostering a positive attitude of students to knowledge,
  • instill confidence in students by means of a lesson.
• Educational:
  • development of analytical thinking,
  • development of "skills to learn": to use knowledge, skills and abilities in the educational process,
  • development of logical thinking, the ability to clearly articulate their thoughts.

Equipment: interactive board, presentation, cards.

DURING THE CLASSES

I. Organizational moment

- Today in the lesson we will remember the definitions of right-angled, isosceles, equilateral triangles. Let's repeat the properties of the angles of triangles. Using the properties of internal one-sided and internal cross-lying angles, we will prove the theorem on the sum of the angles of a triangle and learn how to apply it in solving problems.

II. Orally(Slide 2)

1) Find right-angled, isosceles, equilateral triangles in the figures.
2) Define these triangles.
3) Formulate the properties of the angles of an equilateral and isosceles triangle.

4) In the figure KE II NH. (slide 3)

– Specify secants for these lines
– Find internal one-sided angles, internal cross-lying angles, name their properties

III. Explanation of new material

Theorem. The sum of the angles of a triangle is 180 o

According to the formulation of the theorem, the guys build a drawing, write down the condition, conclusion. Answering the questions, independently prove the theorem.

Given: Prove:

Proof:

1. Draw a line BD II AC through the vertex B of the triangle.
2. Specify secants for parallel lines.
3. What can be said about the angles CBD and ACB? (make a record)
4. What do we know about angles CAB and ABD? (make a record)
5. Replace angle CBD with angle ACB
6. Make a conclusion.

IV. Finish the offer.(Slide 4)

1. The sum of the angles of a triangle is ...
2. In a triangle, one of the angles is equal, the other, the third angle of the triangle is equal to ...
3. The sum of the acute angles of a right triangle is ...
4. The angles of an isosceles right triangle are equal to ...
5. The angles of an equilateral triangle are equal ...
6. If the angle between the sides of an isosceles triangle is 1000, then the angles at the base are ...

V. A bit of history.(Slides 5-7)

Proof of the theorem on the sum of angles of a triangle "The sum of the interior the angles of a triangle are equal to two right angles" attributed to Pythagoras (580-500 BC)
Ancient Greek scholar Proclus (410-485 AD),