Definition of triangle properties and deterministic theorems

Isosceles triangle:

Definition: A triangle with two equal sides is an isosceles triangle. In an isosceles triangle, the two sides that are equal are called the waist, the other side is called the bottom edge, the angle between the two waists is called the top angle, and the angle between the waist and the bottom edge is called the bottom angle.

Properties:1The two waists of an isosceles triangle are equal; 2.

The two base angles of an isosceles triangle are equal; 3.An isosceles triangle is an axisymmetric figure; 4.The bisector of the top angle of the isosceles triangle, the middle line on the bottom edge, and the high coincidence on the bottom edge are all axes of symmetry of the isosceles triangle.

Verdict: 1A triangle with two equal sides is an isosceles triangle; 2.If a triangle has two equal angles, then the sides opposite each other are also equal.

Equilateral triangles:

Definition: A triangle with equal three sides is an equilateral triangle, also called a regular triangle.

Properties:1An equilateral triangle is an axisymmetric figure with three axes of symmetry, and the perpendicular bisector of any side is its axis of symmetry; 2.The three angles of an equilateral triangle are all equal, and each angle is 60°.

Verdict: 1A triangle with three equal sides is an equilateral triangle; 2.There is an isosceles triangle with an angle of 60° that is an equilateral triangle; 3.There are two triangles with an angle of 60° that are equilateral triangles.

Right Triangle:

Definition: A triangle with an inside angle that is a right angle is called a right triangle. Among them, the two sides that make up the right angle are called the right angled edge, and the side opposite the right angle edge is called the hypotenuse.

Properties:1The two coangles of a right triangle are interconnected; 2.The middle line on the hypotenuse of a right triangle is equal to half of the hypotenuse; 3.In a right-angled triangle, the right-angled side opposite by a 30° angle is equal to half of the hypotenuse; 4.Pythagorean theorem.

Verdict: 1A triangle with an angle that is a right angle is a right triangle; 2.

A triangle with two angles remaining to each other is a right triangle; 3.If the middle line on one side of a triangle is equal to half of the side, then the triangle is a right-angled triangle; 4.If the three sides of a triangle are long a, b, and c satisfies a 2 + b 2 = c 2, then the triangle is a right triangle.

Isosceles triangle: A triangle with two equal sides is an isosceles triangle.

Equilateral triangles: Triangles with equal three sides are equilateral triangles (equilateral triangles are special isosceles triangles).

Right Triangle: A triangle with a right angle is called a right triangle.

Properties: The two sides of an isosceles triangle are equal, and the corresponding two angles are equal.

An equilateral triangle has three equal sides and three equal angles.

The two acute angles of a right triangle complement each other.

Decision theorem: A triangle with two equal sides (two corners) is an isosceles triangle.

A triangle where all three sides (three corners) are equal is an equilateral triangle.

A triangle with an angle that is a right angle is a right triangle;

Basic properties of triangles:

Property 1: The sum of the two sides of the triangle is greater than the third side; The difference between the two sides is less than the third side. (Relationship of Triangle Edges).

Property 2: The sum of the three inner angles of a triangle is equal to 180° (the relationship between the three inner angles).

Property 3: Triangles have stability.

The triangle theorem is as follows:

1. The sum of the inner angles of the triangle on the plane is equal to 180° (the sum of the internal angles theorem).

2. The outer angle of the triangle on the plane.

The sum is equal to 360° (the sum of the outer angles theorem).

3. On a plane, the outer angles of a triangle are equal to the sum of the two inner angles that are not adjacent to it.

Corollary: One outer angle of a triangle is greater than any of its inner angles that are not adjacent to it.

4. Sail At least two of the three inner angles of a triangle are acute.

5. At least one angle in the triangle is greater than or equal to 60 degrees, and at least one angle is less than or equal to 60 degrees.

Similar triangles:

1.The three sides of a triangle are proportional to the three sides of another triangle, so the two triangles are similar.

Abbreviation: three sides correspond to two proportional triangles similar).

2.If the two sides of a triangle correspond to the two sides of another triangle and the angles are equal, then the two triangles are similar (for short: the two sides correspond to the two triangles of the hail that are proportional and have equal angles).

3.If the two corners of a triangle correspond equally to the two angles of another triangle, then the two triangles are similar (abbreviation: two angles correspond to two triangles that are equal).

4.If a right triangle.

The hypotenuse and a right-angled edge are proportional to the hypotenuse and a right-angled side of another right-angled triangle, then the two triangles are similar.

There are 5 theorems for determining the congruence of triangles. 1. A triangle with three sides corresponding to equality is a congruent triangle. SSS (edge edge) 2, the two sides and their angles correspond to the triangle that is equal to the digging suspicion is a congruent triangle.

SAS (Corner Edge) 3, the two corners and their edges correspond to the equal congruence of triangles. ASA (Corner Corner) 4, two corners and the opposite side of one of the corners correspond to equal triangle congruence. AAS (Corner Edge) 5, in a pair of right triangles, the hypotenuse and the other right angle side are equal.

rhs (right angle, hypotenuse, edge).

Triangle congruence slips smoothly: congruent triangle, the nature of which should be clarified. The corresponding edges are equal, and the corresponding angles are the same. Corners, corners, edges, edges, corners, four theorems to be memorized.

Triangle Determination Method 1:

1. Acute triangle: The three inner angles of the triangle are less than 90 degrees.

2. Right triangle: one of the three inner angles of the triangle is equal to 90 degrees, which can be recorded as RT.

3. Obtuse triangle: One of the three inner angles of the triangle is greater than 90 degrees of collapse.

Three, the judgment of the shirt hand angle shape judgment method two:

1. Acute triangle: The maximum angle of the three inner angles of the triangle is less than 90 degrees.

2. Right triangle: The maximum angle of the three inner angles of the triangle is equal to 90 degrees.

3. Obtuse triangle: The maximum angle of the three inner angles of the triangle is greater than 90 degrees and less than 180 degrees.

Properties: 1 The sum of the two sides of the triangle must be greater than the third side, so it can also be proved that the difference between the two sides of the triangle must be less than the third side.

2 The sum of the inner angles of the triangle is equal to 180 degrees.

3 The bisector of the top angle of the isosceles triangle, the middle line of the bottom edge, and the high overlap of the bottom edge, that is, the three lines are one.

4 The sum of the squares of the two right-angled sides of a right-angled triangle is equal to the square of the hypotenuse - the Pythagorean theorem. The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse.

5 The outer angles of a triangle are equal to the sum of the two inner angles that are not adjacent to it.

6 A triangle has at least 2 acute angles out of the 3 inner angles.

7 The three angular bisectors of the triangle intersect at one point, the straight line where the three high lines are located intersect at one point, and the three middle lines meet at one point.

8 The ratio of the area of a triangle with equal bases is equal to the ratio of its height, and the ratio of the area of a triangle of equal height is equal to the ratio of its base.

Determination of similar triangles:

1) The three sides correspond to two triangles that are proportional to each other.

2) The two sides correspond to two triangles that are proportional and their angles are similar.

3) The angles correspond to two equal triangles.

4) If the hypotenuse and one right-angled side of a right-angled triangle correspond to the hypotenuse and one right-angled side of another right-angled triangle, then the two triangles are similar.

The left and right triangles are congruent and equal in area. The upper and lower triangles are similar, and the ratio of the area of the sock is equal to the square of the upper bottom than the square of the bottom bottom.

Let s1=ms

Then: s2=ns, s3=(s2)*(n m)=(n 2 m)ss4=(s3)*(m n)=ns

So: s1:s2:

s3:s4=m:n:

n^2/m):ns1:s2:

s3:s4=(m^2) :mn :

n 2) :mn judgment method: 1, acute triangle:

The largest of the three inner angles of the collapse induced triangle is less than 90 degrees.

2. Right triangle: The maximum angle of the three inner angles of the triangle is equal to 90 degrees.

3. Obtuse triangle: The maximum angle of the three inner angles of the triangle is greater than 90 degrees, which is less than 180 degrees.

Among them, acute triangles and obtuse triangles are collectively called oblique triangles.