2. The inverse theorem of Pythagorean theorem: If the lengths of triangle A, B and C satisfy A2+B2 = C2. Then this triangle is a right triangle.
A proposition that is proved to be correct is called a theorem.
We call two propositions with opposite topics and conclusions reciprocal propositions. If one of them is called the original proposition, then the other is called its inverse proposition. (Example: Pythagorean Theorem and Pythagorean Theorem Inverse Theorem)
4. The nature of right triangle
(1), the two acute angles of a right triangle are complementary. It can be expressed as follows: ∠ C = 90 ∠ A+∠ B = 90.
(2) In a right-angled triangle, the right-angled side facing an angle of 30 is equal to half of the hypotenuse.
∠A=30
It can be expressed as follows: BC= AB
∠C = 90°
(3) The median line on the hypotenuse of a right triangle is equal to half of the hypotenuse.
∠ACB=90
It can be expressed as follows: CD= AB=BD=AD.
D is the midpoint of AB.
5, photography theorem
In a right-angled triangle, the high line on the hypotenuse is the proportional average of two right-angled sides on the hypotenuse, and each right-angled side is their photographic proportional average on the hypotenuse and hypotenuse.
∠ACB=90
CD⊥AB
6. Common relational expressions
It can be obtained from the triangle area formula: AB CD=AC BC.
7. Determination of right triangle
1. A triangle with right angles is a right triangle.
2. If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.
3. Inverse theorem of Pythagorean theorem: If three sides of a triangle are related, then the triangle is a right triangle.
8. Propositions, theorems and proofs
1, the concept of proposition
A statement that judges a thing is called a proposition.
Understanding: The definition of a proposition contains two meanings:
(1) The proposition must be a complete sentence;
(2) What must be judged in this sentence.
2. Classification of propositions (according to right or wrong)
True proposition (correct proposition)
proposition
False position
The so-called correct proposition is: if the topic is established, then the conclusion must be established.
The so-called wrong proposition is: if the topic is established, it cannot be proved that the conclusion is always established.
3. Axiom
The true proposition that people have summed up in long-term practice and recognized by people is called axiom.
4. theorem
A proposition that is judged to be correct by reasoning is called a theorem.
Step 5 prove
The reasoning process of judging the correctness of a proposition is called proof.
6, the general steps of proof
(1) Draw a picture according to the meaning of the question.
(2) According to the topic, conclusion, combined with graphics, write what is known and verified.
(3) Through analysis, find out the proof method of known derivation and write the proof process.
9, the center line in the triangle
The line segment connecting the midpoints of two sides of a triangle is called the midline of the triangle.
(1) Triangle * * * has three median lines, which form a new triangle.
(2) The midline of a triangle can be distinguished.
The midline theorem of triangle: the midline of triangle is parallel to the third side and equal to half of it.
The function of triangle midline theorem;
Position relation: It can be proved that two straight lines are parallel.
Quantitative relationship: it can prove the doubling relationship of line segments.
Common conclusion: Any triangle has three median lines, from which there are:
Conclusion 1: Three median lines form a triangle, and its circumference is half that of the original triangle.
Conclusion 2: Three median lines divide the original triangle into four congruent triangles.
Conclusion 3: Three median lines divide the original triangle into three parallelograms with equal areas.
Conclusion 4: A midline of the triangle is equally divided with the intersecting midline.
Conclusion 5: The included angle between any two median lines in a triangle is equal to the vertex angle of the triangle corresponding to this included angle.
10 mathematical formula.
Square difference formula: There are two items in the square difference formula. Remember that the symbols are opposite, multiply the beginning and end by the end, and don't confuse it with the complete formula.
Complete square formula: there are three complete squares, the first and last symbols are fellow villagers, the first and last squares, and the first and second squares are placed in the middle; The first and last brackets are square, and the symbol of the last item follows the center.
quadrilateral
1. The theorem of the sum of inner and outer angles of quadrilateral;
The sum of the internal angles of (1) quadrilateral is equal to 360;
(2) The sum of the external angles of the quadrilateral is equal to 360.
2. The theorem of the sum of inner and outer angles of polygons;
(1) The sum of the internal angles of n polygons is equal to (n-2)180;
(2) The sum of the external angles of any polygon is equal to 360.
3. The properties of parallelogram:
Because ABCD is a parallelogram?
4. Determination of parallelogram:
5. The nature of the rectangle:
Because ABCD is rectangular?
6. Determination of rectangle:
? The quadrilateral ABCD is a rectangle.
7. The nature of diamonds:
Because ABCD is a diamond.
8. The Diamond Trial:
? The quadrilateral ABCD is a diamond.
9. The nature of the square:
Because ABCD is a square.
( 1) (2)(3)
10. Square:
? The quadrilateral ABCD is a square.
(3) ABCD is rectangular.
Once again ∵AD=AB
∴ quadrilateral ABCD is a square
1 1. Characteristics of isosceles trapezoid;
Because ABCD is isosceles trapezoid?
12. Determination of isosceles trapezoid;
? The quadrilateral ABCD is an isosceles trapezoid.
(3)∫ABCD is trapezoidal and AD∨BC.
∵AC=BD∴ABCD quadrilateral is isosceles trapezoid.
14. triangle median theorem;
The center line of a triangle is parallel to the third side and equal to half of it.
15. Trapezoidal midline theorem;
The center line of the trapezoid is parallel to the two bottom sides and equal to half of the sum of the two bottom sides.
A basic concept: quadrilateral, its inner angle, its outer angle, the distance between polygon and parallel line, parallelogram, rectangle, diamond, square, central symmetry, central symmetry figure, trapezoid, isosceles trapezoid, right-angled trapezoid, triangle midline and trapezoid midline.
Two theorems: the related theorem of central symmetry
1. The congruence of two graphs with central symmetry. ※ 。
2. For two graphs with symmetrical centers, the connecting line of symmetrical points passes through the symmetrical center and is equally divided by the symmetrical center. ※ 。
3. If the line connecting the corresponding points of two graphs passes through a certain point and is equally divided by the point, then the two graphs are symmetrical about the point. ※ 。
Three formulas:
1.s diamond = ab=ch. (A and B are diagonal lines of the diamond, C is the side length of the diamond, and H is the height on the side of C)
2.s parallelogram = ah. A is the side length of the parallelogram, and h is the height on a).
3.s trapezoid = (a+b)h=Lh. (A and B are trapezoidal bottom, H is trapezoidal height, and L is trapezoidal center line)
Four common sense:
1. If n is the number of sides of a polygon, the diagonal number formula is: ※ 。
2. Regular graph folding generally "gives a pair of congruences and a pair of similarities".
3. As shown in the figure: subordination of parallelogram, rectangle, diamond and square.
4. Common only axisymmetric figures are: angle, isosceles triangle, equilateral triangle, equilateral polygon, isosceles trapezoid ...; Only the figures with central symmetry are: parallelogram ...; Bisymmetric figures include: line segments, rectangles, diamonds, squares, even polygons, circles ... Note: line segments have two symmetry axes.
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