I. Axiom (without proof)
1. Two lines are cut by the third line. If congruent angles are equal, then two straight lines are parallel;
2. Two parallel lines are cut by a third straight line, and the same angle is equal;
3. The included angle between two sides corresponds to the coincidence of two equal triangles; (SAS)
4. The angle and its clamping edge correspond to the coincidence of two triangles; (ASA)
5. A triangle with two equal sides is congruent; (SSS)
6. Equilateral triangles have equal sides and equal angles.
7. Axiom of line segment: between two points, the line segment is the shortest.
8. axiom of straight line: there is only one straight line after two o'clock.
9. Parallelism axiom: One and only one point outside the straight line is parallel to the known straight line.
10. verticality: after passing through a point on the outside or straight line, one and only one straight line is perpendicular to the known straight line.
Note: (1) Among them, 1-6 can be used as the basis for proving other theorems, and 7- 10 should be understood as a basic fact.
(2) The related properties of equality and inequality can also be regarded as axioms.
Axioms and theorems learned in junior high school are classified as follows:
First, straight lines and angles
Between 1 and two points, the segment is the shortest.
There is a straight line after two o'clock, and there is only one straight line.
3. The complementary angles of the same angle or equal angle are equal, and the complementary angles of the same angle or equal angle are equal.
4, the vertex angles are equal
Second, parallel and vertical.
5. After passing a point outside or on a straight line, one and only one straight line is perpendicular to the known straight line.
6. After a point outside the straight line is known, one and only one straight line is parallel to the known straight line.
7. Of all the line segments connecting points outside the straight line and points on the straight line, the vertical line segment is the shortest.
8. The parallel lines sandwiched between two parallel lines are equal.
9. Determination of parallel lines:
(1) Same angle, two straight lines are parallel;
(2) The internal dislocation angles are equal and the two straight lines are parallel;
(3) The internal angles on the same side are complementary and the two straight lines are parallel;
(4) Two straight lines perpendicular to the same straight line are parallel to each other.
(5) If two straight lines are parallel to the third straight line, then the two straight lines are also parallel.
(6) Using the midline theorem of triangle
10, properties of parallel lines:
(1) Two straight lines are parallel with the same included angle.
(2) The two straight lines are parallel and the internal dislocation angles are equal.
(3) Two straight lines are parallel and complementary.
Three, the angle bisector, perpendicular bisector, the change of the figure (axis symmetry, rotation).
1 1, the nature of the bisector: the distance from the point on the bisector to both sides of the corner is equal.
12. Judgment of the bisector of an angle: the point where two sides of an angle are equidistant is on the bisector of this angle.
13. The nature of the vertical line in the line segment: the distance between a point on the vertical line in the line segment and the two endpoints of the line segment is equal.
14. Determination of the vertical line in a line segment: the point with equal distance to the two endpoints of a line segment is on the vertical line in this line segment.
15, the essence of axial symmetry:
(1) If the graph is symmetrical about a straight line, then the line segments connecting the corresponding points are vertically bisected by the symmetry axis.
(2) The corresponding line segments are equal and the corresponding angles are equal.
16. Translation: After translation, every point on the map moves the same distance in the same direction. After translation, the shape and size of the new image and the original image have not changed, that is, the congruent image. That is, the corresponding line segments are parallel and equal, the corresponding angles are equal, and the line segments connected by the corresponding points are parallel and equal.
17, rotational symmetry:
(1) Every point in the graph rotates by the same angle around the center of rotation.
(2) The distance from the corresponding point to the rotation center is equal;
(3) The corresponding line segments are equal and the corresponding angles are equal.
18, center symmetry:
(1) has all the properties of rotational symmetry:
(2) The line segments connected by each pair of corresponding points on the central symmetric figure are equally divided by the symmetric center.
Fourth, triangle.
(1) general properties
19, triangle interior angle sum theorem: triangle interior angle sum is equal to 180.
20. The nature of the external angle of a triangle: ① One external angle of a triangle is equal to the sum of two internal angles that are not adjacent to it; (2) The outer angle of a triangle is larger than any inner angle that is not adjacent to it; ③ The sum of the external angles of the triangle is equal to 360.
2 1, trilateral relationship: (1) The sum of two sides is greater than the third side; (2) The difference between the two sides is smaller than the third side.
22. Triangle midline theorem: the midline of a triangle is parallel to the third side and equal to half of the third side.
23. The perpendicular lines of the three sides of a triangle intersect at a point (the outer center), and the distance from the point to the three vertices (the radius of the circumscribed circle) is equal.
24. The three bisectors of a triangle intersect at a point (the heart), and the distance from the point to the three sides (the radius of the inscribed circle) is equal.
(2) Special properties:
25, isosceles triangle, equilateral triangle
(1) The two base angles of an isosceles triangle are equal.
(2) If the two angles of a triangle are equal, then the opposite sides of the two angles are also equal.
(3) Theorem of "Three Lines in One": The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other.
(4) The three internal angles of an equilateral triangle are all equal, and each internal angle is equal to 60.
(5) A triangle with three equal angles is an equilateral triangle.
(6) An isosceles triangle with an angle of 60 is an equilateral triangle.
26, right triangle:
(1) The two acute angles of a right triangle are complementary;
(2) Pythagorean theorem: the sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse;
(3) Pythagorean Inverse Theorem: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then this triangle is a right triangle.
(4) The median line on the hypotenuse of a right triangle is equal to half of the hypotenuse.
(5) In a right-angled triangle, if an acute angle is equal to 30, then the right-angled side it faces is equal to half of the hypotenuse.
(6) The median line of one side of a triangle is equal to half of this side. This triangle is a right triangle.
Five, quadrilateral
27. Related axioms and theorems in polygons:
The sum of the internal angles of the (1) quadrilateral is 360.
⑵ Theorem of the sum of internal angles of polygons: the sum of internal angles of n polygons is equal to (n-2) × 180.
⑶ Theorem of the sum of external angles of polygons: the sum of external angles of any polygon is 360.
28, the nature of the parallelogram:
(1) The opposite sides of the parallelogram are parallel and equal;
(2) The diagonals of the parallelogram are equal;
(3) The diagonal of the parallelogram is equally divided.
29, parallelogram judgment:
(1) Two groups of parallelograms with parallel opposite sides are parallelograms;
(2) A group of quadrilaterals with parallel and equal opposite sides are parallelograms;
(3) Two groups of quadrangles with equal opposite sides are parallelograms;
(4) Two groups of quadrangles with equal diagonal are parallelograms;
(5) The quadrilateral whose diagonal lines bisect each other is a parallelogram.
30, the nature of the rectangle:
(1) has all the properties of a parallelogram.
(2) All four corners of a rectangle are right angles;
(3) The diagonals of the rectangle are equal and equally divided.
3 1, rectangular determination:
(1) A parallelogram with right angles is a rectangle.
(2) A quadrilateral with three right angles is a rectangle.
(3) Parallelograms with equal diagonals are rectangles.
32, the nature of the diamond:
(1) has all the properties of a parallelogram.
(2) All four sides of the diamond are equal;
(3) The diagonal lines of the diamond are divided vertically, and each diagonal line divides a set of diagonal lines equally.
33, diamond judgment:
(1) A quadrilateral with four equilateral sides is a diamond.
(2) A set of parallelograms with equal adjacent sides is a rhombus.
(3) Parallelograms with diagonal lines perpendicular to each other are rhombic.
34, the nature of the square:
(1) has all the properties of rectangle and diamond.
(2) All four corners of a square are right angles;
(3) All four sides of a square are equal;
(4) The two diagonals of a square are equal and bisected vertically, and each diagonal bisects a set of diagonals.