1. Between the straight line and the angle 1 and two points, the line segment is the shortest. 2. There is a straight line passing through two points, and there is only one straight line. 3. The vertex angles are equal. The complementary angle (or complementary angle) of the same angle is equal; The complementary angles (or complementary angles) of equal angles are equal. 4. Crossing a point outside or on a straight line, there is one and only one straight line perpendicular to the known straight line. 5.( 1) There is one and only one straight line parallel to the known straight line through a point outside the known straight line. (2) If two straight lines are parallel to the third straight line, then the two straight lines are also parallel. 6. Determination of parallel lines: (1) Two lines are parallel (2), with the same offset angle, and two lines are parallel (3), with complementary internal angles and parallel lines. 7. Characteristics of parallel lines: (1), two straight lines are parallel and have the same angle. ② The internal dislocation angles are equal. (3) Two straight lines are parallel. Complementarity of ipsilateral internal angle 8. The nature of the bisector: the distance from the point on the bisector to both sides of the angle is equal. Judgment of the bisector of the angle: the points with equal distance on both sides of the angle are on the bisector of the angle. 9. The nature of the vertical line in the line segment: the distance from the point on the vertical line in the line segment to the two endpoints of the line segment is equal. Judgment of the perpendicular line of a line segment: to the point where the two endpoints of a line segment are at the same distance, on the perpendicular line of this line segment, 2. Related axioms and theorems in triangle, polygon 10, triangle: (1) 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) An outer angle of a triangle is larger than any inner angle that is not adjacent to it; ③ Theorem that the sum of the external angles of a triangle is equal to 360 (2): The sum of the internal angles of a triangle is equal to 180 (3) The sum of any two sides of a triangle is greater than the third side. 4) midline theorem of triangle: the midline of triangle is parallel to the third side, And equal to half of the third side 1 1, related axioms and theorems in polygons: (1) The sum of internal angles of polygons: the sum of internal angles of n polygons is equal to (n-2) × 180 (2) The sum of external angles of polygons: the sum of external angles of any polygon is then. If a triangle has two equal angles, the opposite sides of the two angles are also equal. (abbreviated as "equilateral") (3) The theorem of "three lines in one" of isosceles triangle: the bisector of the top angle of isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide with each other, which is called "three lines in one" for short. (4) All internal angles of an equilateral triangle are equal, and each internal angle is equal to 60 (5) three. An isosceles triangle with an angle equal to 600 is an equilateral triangle; Triangles with three equal angles are equilateral triangles (14) and right triangles (1). The two acute angles of a right triangle are complementary. (2) Pythagorean theorem: The sum of squares of two right sides 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 triangle, if an acute angle is equal to 30, then the right side it faces is equal to half of the hypotenuse. 3. Special quadrilateral 15. Properties of parallelogram: (1) The opposite sides of parallelogram are parallel and equal. (2) The diagonals of parallelograms are equal. (3) The diagonals of the parallelogram are equally divided. (3) .5438+06, parallelogram judgment: (1) Two groups of parallelograms with parallel opposite sides are parallelograms (2) Two groups of parallelograms with equal opposite sides are parallelograms (3) (4) Two groups of parallelograms with equal diagonal angles are parallelograms (5) Parallelograms with diagonal bisection are parallelograms/kloc-0. (2) The diagonal lines of the rectangle are equal and equal to each other 19. Judgment of rectangle: (1) A parallelogram with one right angle is a rectangle (2) A quadrilateral with three right angles is a rectangle (3) A parallelogram with equal diagonal lines is a rectangle 20. The nature of the diamond: (1) All four sides of the diamond are equal; (2) The diagonal of the diamond is divided vertically. And each diagonal bisects a set of diagonal 2 1 and rhombus: (1) A set of parallelograms with equal adjacent sides is rhombus; (2) A quadrilateral with equal four sides is rhombus; (3) Parallelograms with mutually perpendicular diagonals are rhombus; the nature of a square: (1) All four corners of a square are right angles; (2) All four sides of a square are equal; (3) The two diagonals of a square are equal and equally divided vertically. The judgment that each diagonal bisects a set of diagonal lines 23 and squares: (1) A diamond with a right angle is a square; (2) A group of rectangles with equal adjacent sides are squares; (3) Two rectangles with vertical diagonal lines are squares; (4) The rhombus with equal diagonal lines is a regular trapezoid; One set of quadrangles with parallel opposite sides and another set of non-parallel sides is trapezoid 24; Determination of isosceles trapezium: (1) Two trapeziums with equal internal angles on the same base are isosceles trapezium (2) Two trapeziums with equal diagonal lines are isosceles trapezium (25) Properties of isosceles trapezium: (1) Two internal angles on the same base are equal (2) Two diagonal lines of isosceles trapezium are equal (26), and the center line of the trapezoid is parallel to the two trapezium. It is equal to half of the sum of two cardinality. 4. Similar shapes and consistency. Properties of similar polygons: (1) The sides corresponding to similar polygons are proportional; (2) The angles corresponding to similar polygons are equal; (3) The ratio of the perimeters of similar polygons is equal to the similarity ratio; (4) The area ratio of similar polygons is equal to the square of the similarity ratio; (5) The corresponding angles of similar triangles are equal, and the corresponding edges are proportional; Similar triangles's ratio corresponding to high and the ratio corresponding to the center line are all equal to the similar ratio; The ratio of similar triangles perimeter is equal to similarity ratio; The area ratio of similar triangles is equal to the square of the similarity ratio. 28. similar triangles's judgment: (1) If two angles of a triangle are equal to two angles of another triangle, then the two triangles are similar. (2) If two sides of a triangle are proportional to two sides of another triangle, and the included angles are equal, then the two triangles are similar. (3) If three sides of a triangle are proportional to three sides of another triangle, then the two triangles are similar. 29. The corresponding edges and angles of congruent polygons are equal. 30. congruent triangles's judgment: (1) If three sides of two triangles are equal, they are congruent (S.S.) (2) If two triangles have two sides and their included angles are equal respectively, they are congruent (S.A.S.) (3) If two angles of two triangles and their sandwiched sides are equal respectively. Then these two triangles are congruent (A.S.A.) (4) with two angles and the opposite side of one angle corresponds to two equal triangles respectively (A.A.S.) (5) If the hypotenuse of two right-angled triangles corresponds to a right-angled side, then these two right-angled triangles are congruent (H.L.) V. Circle 365438+. (2) The circumferential angles of semicircles or diameters are all equal, equal to 90 (right angle); (3) A chord with a circumferential angle of 90 is the diameter of a circle. 32. In the same circle, the circumferential angle of the same or equal arc is equal, which is equal to half of the central angle of the arc; The arc subtended by equal circumferential angles is equal to 33; Three points that are not on the same straight line define a circle 34; (1) The straight line passing through the outer end of the radius and perpendicular to this radius is the tangent of the circle; (2) The tangent of the circle is perpendicular to the radius of the tangent point 35; Two tangents that can be drawn from a point outside the circle have the same tangent length; The line between this point and the center bisects the included angle of these two tangents 36. The diameter (not the diameter) of the bisecting chord is perpendicular to the chord and bisects the two arcs opposite to the chord. 6. transformation 37. Axisymmetric: (1) congruence of two figures symmetric about a line; If two figures are symmetrical about a straight line, then the symmetry axis is the middle vertical line connecting the corresponding points; (2) Two figures are symmetrical about a straight line. If their corresponding line segments (or extension lines) intersect, the intersection must be on the axis of symmetry; (3) Two figures are symmetrical about a straight line. If their corresponding line segments (or extension lines) intersect, the intersection must be on the axis of symmetry; (4) If the line connecting the corresponding points of two graphs is vertically bisected by the same straight line, the two graphs are symmetrical about this straight line 38. Translation: (1) Translation does not change the shape and size of the graphics (that is, the two graphics are identical before and after translation); (2) The corresponding line segments are parallel and equal (or on the same straight line), and the corresponding angles are equal; (3) After translation, the line segments corresponding to two points are parallel (or on the same straight line) and equal. 39. Rotation: (1) Rotation does not change the shape and size of the graph (that is, the two graphs are the same before and after rotation) (2) The angles formed by the connecting line of any pair of corresponding points and the rotation center are equal (both are rotation angles) (3) After rotation, they correspond. (2) For two graphs with symmetrical centers, the straight line connecting the symmetrical points passes through the symmetrical centers; (3) If a straight 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 4 1 and similar: (1) If the two graphs are not only similar, but also the straight lines of each group of corresponding vertices pass through the same point, then such two graphs are called similar graphs, and this point is called similar center, and the similarity ratio at this time is again. (2) The ratio of the distance between any pair of corresponding points on the similarity graph and the similarity center is equal to the similarity ratio.