Whether the outer angle of the triangle is greater. The external angle of an arbitrary triangle is greater than each internal angle not adjacent to it. Proof. Let ABC be an arbitrary triangle. Consider

The external angle of an arbitrary triangle is greater than each internal angle not adjacent to it. Proof. Let ABC be an arbitrary triangle. Consider, for example, the exterior angle BCD and prove that it is greater than the interior angle ABC. To do this, draw a straight line through the vertex A and the midpoint E of the side BC and draw a segment EF equal to AE on it. Triangles ABE and FCE are equal according to the first sign of equality of triangles (BE = CE, AE = FE, AEB = FEC). Hence ABC = BCF. But the vertex F lies inside the angle BCD. Therefore angle BCF is only part of angle BCD. So BCD > ABC. ABC.">

In an arbitrary triangle, the larger angle lies opposite the larger side. Proof. Let side AB be greater than side AC in triangle ABC. Let us prove that angle C is greater than angle B. To do this, we plot a segment AD on the ray AB, equal to the side AC. Triangle ACD is isosceles. Therefore, 1 = 2. Angle 1 is part of angle C. Therefore, 1 B. Therefore, we have C > 1 = 2 > B. 1 = 2 > B.">

It is known that in the triangle ABC BC > AC > AB. Which of the angles is larger: a) B or A; b) C or A; c) B or C? Answer: a), b) A; c) B. A C B AC>AB. Which of the angles is larger: a) B or A; b) C or A; c) B or C? Answer: a), b) A; c) B. A C B "> AC> AB. Which of the angles is greater: a) B or A; b) C or A; c) B or C? Answer: a), b) A; c) B. A C B"> AC> AB. Which of the angles is larger: a) B or A; b) C or A; c) B or C? Answer: a), b) A; c) B. A C B" title=" It is known that in the triangle ABC BC > AC > AB. Which of the angles is greater: a) B or A; b) C or A; c) B or C? Answer: a), b) A; c) B. A C B"> title="It is known that in the triangle ABC BC > AC > AB. Which of the angles is larger: a) B or A; b) C or A; c) B or C? Answer: a), b) A; c) B. A C B"> !}

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Answer: a) BC > AC > AB; Compare the sides of triangle ABC if: a) A > B > C; b) A > B, B = C. b) BC > AB, AC = AB. AC > AB; Compare the sides of triangle ABC if: a) A > B > C; b) A > B, B = C. b) BC > AB, AC = AB."> AC > AB; Compare the sides of triangle ABC if: a) A > B > C; b) A > B, B = C b) BC > AB, AC = AB."> AC > AB; Compare the sides of triangle ABC if: a) A > B > C; b) A > B, B = C. b) BC > AB, AC = AB." title=" Answer: a) BC > AC > AB; Compare the sides of triangle ABC if: a) A > B > C; b) A > B, B = C. b) BC > AB, AC = AB."> title="Answer: a) BC > AC > AB; Compare the sides of triangle ABC if: a) A > B > C; b) A > B, B = C. b) BC > AB, AC = AB."> !}

In triangle ABC, the inequality AC > BC is satisfied, CD is the median. Which angle is greater than ACD or BCD? Answer: BCD. A B C D BC, CD - median. Which angle is greater than ACD or BCD? Answer: BCD. A B C D "> BC, CD is the median. Which of the angles is greater than ACD or BCD? Answer: BCD. A B C D"> BC, CD is the median. Which angle is greater than ACD or BCD? Answer: BCD. A B C D" title=" In triangle ABC, the inequality AC > BC, CD is the median. Which of the angles is greater than ACD or BCD? Answer: BCD. A B C D"> title="In triangle ABC, the inequality AC > BC is satisfied, CD is the median. Which angle is greater than ACD or BCD? Answer: BCD. A B C D"> !}

slide 1

An external angle of an arbitrary triangle is greater than any internal angle not adjacent to it. Proof. Let ABC be an arbitrary triangle. Consider, for example, the exterior angle BCD and prove that it is greater than the interior angle ABC. To do this, draw a straight line through the vertex A and the midpoint E of the side BC and draw a segment EF equal to AE on it. Triangles ABE and FCE are equal according to the first sign of equality of triangles (BE = CE, AE = FE, AEB = FEC). Hence ABC = BCF. But the vertex F lies inside the angle BCD. Therefore angle BCF is only part of angle BCD. So BCD > ABC.

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Theorem 2

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Exercise 1

Can the exterior angle of a triangle be equal to its interior angle? Answer: Yes, in a right triangle.

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Exercise 2

Can the exterior angle of a triangle be less than its interior angle? Answer: Yes, in an obtuse triangle.

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Exercise 3

How many triangles can have: a) right angles; b) obtuse corners? Answer: a) b) One.

Slide 6

Exercise 4

It is known that in the triangle ABCBC > AC > AB. Which of the angles is larger: a) B or A; b) C or A; c) B or C? Answer: a), b) A; c) b.

Slide 7

Exercise 5

In triangle ABC, side AB is the longest. What are the acute angles of this triangle? What can be angle C? Answer: Angles A and B are acute. Angle C can be acute, right or obtuse.

Slide 8

Exercise 6

Figure 1 BC.

Slide 9

Exercise 7

Is it true that in an arbitrary triangle the larger side lies opposite the larger angle? Answer: Yes.

Slide 10

Exercise 8

Answer: a) BC > AC > AB; Compare sides of triangle ABC if: a) A>B>C; b) A > B, B = C. b) BC > AB, AC = AB.

Isosceles triangle and its properties. middle line triangle.

TRIANGLE - geometric figure, consisting of three points (vertices) and three segments (sides) connecting them in pairs. The sides of a triangle are often denoted by small letters, which correspond to capital letters denoting opposite vertices.

The sum of the lengths of all sides of a triangle is called the perimeter.

Triangles differ in the size of the angles: acute, rectangular, obtuse. If all three angles are acute, then it is an acute triangle. If one of the angles is right, then it is right triangle; one of the angles is obtuse then it is an obtuse triangle.

Three sides of a triangle can be used to determine its shape.

THEOREM. In a triangle, the square of the larger side is equal to the sum of the squares of the other two sides if and only if the triangle is a right triangle.

THEOREM. In a triangle, the square of the longer side is less than the sum of the squares of the other two sides if and only if the triangle is acute.

THEOREM. In a triangle, the square of the larger side is greater than the sum of the squares of the other two sides if and only if the triangle is obtuse.

Triangles differ in the length of the sides: scalene, isosceles, equilateral.

A triangle is isosceles, if two of its sides are equal, these equal sides are called lateral, the third side is called the base of the triangle. A triangle is equilateral if all its sides are equal.

Basic properties of triangles.

THEOREM. In any triangle, the larger side is opposite the larger angle, and vice versa.

Equal angles lie opposite equal sides, and vice versa.

Consequence. All angles in an equilateral triangle are equal.

THEOREM. If two sides of one triangle are respectively equal to two sides of another triangle, then the larger side lies opposite the larger of the angles between them, and the larger angle lies opposite the larger of the remaining sides.

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

Consequence. Each angle in an equilateral triangle is 60º.

Continuing one of the sides of the triangle, we get the external angle

THEOREM. The exterior angle of a triangle is equal to the sum of the interior angles not adjacent to it:

THEOREM. An outer angle of a triangle is greater than every inner angle not adjacent to it.

THEOREM. In a triangle, each side is less than the sum of the other two sides, but greater than their difference.

The median of a triangle drawn from a given vertex is the segment connecting this vertex to the midpoint of the opposite side (the base of the median).

THEOREM. All three medians of the triangle intersect at one point and are divided by this point in a ratio of 2:1, counting from the top. This point of intersection is called the centroid or center of gravity of the triangle.

The height of a triangle drawn from a given vertex is the perpendicular dropped from this vertex to the opposite side or its continuation.

THEOREM. The three heights of a triangle intersect at one point, called the orthocenter of the triangle.

The bisector of a triangle drawn from a given vertex is the segment that connects this vertex to a point on opposite side and bisecting the angle at a given vertex.

THEOREM. The bisectors of a triangle intersect at one point, and this point coincides with the center of the inscribed circle

The middle line of the triangle The line segment that joins the midpoints of two sides of the triangle is called.

THEOREM. The middle line of the triangle is parallel to the base and equal to half of it.

Related information:

I. Organizational moment. 1. The sums of three numbers written along the sides of a triangle have the same value

There are internal and external. What is the outer corner of a triangle? How to find it?

Definition.

External corner of a triangle at a given vertex is the angle with the interior angle of the triangle at that vertex.

How to construct the outer corner of a triangle? We need to extend the side of the triangle.

On the image:

∠3 - external corner at the top A,

∠2 - external corner at the top C,

∠1 - external corner at vertex B.

How many outside angles does a triangle have?

Each vertex of a triangle has two outside corners. To construct an exterior corner at a vertex of a triangle, you can extend either of the two sides on which the given vertex lies. Thus we get 6 external corners.

The external angles of each pair at a given vertex are equal to each other (as ):