I. Lines and angles

Between 1 and two points, the 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. After passing a point outside or on a straight line, one and only one straight line is perpendicular to the known straight line. 5.( 1) After passing a point outside the known straight line, one and only one straight line is parallel to 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) angles are the same, the internal dislocation angles of the two parallel lines (2) are equal, the internal angles of the two parallel lines (3) are complementary, and the two parallel lines (7). Characteristics of parallel lines:

(1) Two straight lines are parallel and have the same angle; (2) Two straight lines are parallel with equal internal angles; (3) Two straight lines are parallel and their internal angles are complementary.

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 an angle: the points at the same distance from both sides of an angle are on the bisector of this 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.

Determination of the vertical line in a line segment: the point with the same distance from the two endpoints of a line segment, the second, triangle and polygon on the vertical line of this line segment.

10, related axioms and theorems in triangle;

The properties of (1) triangle external angles: 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 The sum of the external angles of the triangle is equal to 360.

(2) Theorem of the sum of interior angles of triangle: the sum of interior angles of triangle is equal to 180 (3) The sum of any two sides of triangle is greater than the third side.

(4) Triangle midline theorem: the midline of a triangle is parallel to the third side and equal to half of the third side.

1 1, related axioms and theorems in polygons;

(1) polygon interior angle theorem: the sum of n polygon interior angles is equal to (n-2) × 180 (2) polygon exterior angle theorem: the sum of any polygon exterior angles is 360 (3) Euler formula: number of vertices+number of faces-number of edges =2.

1 2. If the graph is symmetrical about a straight line, then the line segments connecting the corresponding points are vertically bisected by the symmetry axis. 13. Relevant axioms and theorems in isosceles 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) The theorem of "three lines in one" of isosceles triangle: the bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of isosceles triangle coincide, 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.

Quadrilateral (5) The quadrilateral whose diagonal lines bisect each other is a parallelogram 17, and the distance between parallel lines is equal everywhere 18. The properties of the rectangle are:

(1) All four corners of the rectangle are right angles. (2) The diagonal lines of the rectangle are equal and equally divided.

19, rectangle judgment: (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 diagonals is a rectangle 20 and a diamond;

(1) All four sides of the diamond are equal. (2) The diagonal lines of the diamond are divided vertically, and each diagonal line divides a set of diagonal lines equally.

2 1, the judgment of rhombus: (1) A set of parallelograms with equal adjacent sides is a rhombus; (2) A quadrilateral with equal four sides is a rhombus; (3) A parallelogram with orthogonal diagonals is a rhombus 22, which is 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, and each diagonal bisects a set of diagonals 23 and the square:

(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) Two diamonds with equal diagonals are squares.

Trapezoid: the judgment that a set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are trapezoid 24 and isosceles trapezoid;

(1) Two trapeziums with equal inner angles on the same base are isosceles trapeziums (2) Two trapeziums with equal diagonals are isosceles trapeziums.

If an edge is equal to a right-angled edge, then the two right-angled triangles are congruent (H.L.) v.

3 1, (1) In the same circle or equal circle, if one group of two central angles, two arcs and two chords is equal, the corresponding other group is equal; (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 arc or equal arc is equal, which is equal to half of the central angle of the arc; Equal circumferential angles face equal arcs.

33. Three points that are not on the same straight line determine a circle.

34.( 1) The straight line passing through the outer end of the radius and perpendicular to the 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 of a circle can be drawn from a point outside the circle, and their tangents are equal in length. The connecting line between this point and the center of the circle bisects the included angle between the two tangents.

36. The inscribed quadrilateral of a circle is diagonally complementary, and the outer angle is equal to the inner diagonal.

37. Vertical diameter theorem and inference: the diameter perpendicular to the chord bisects the chord and bisects the opposite arc; The diameter (not the diameter) that bisects the chord is perpendicular to the chord and bisects the two arcs opposite the chord.

Sixth, transformation.

37. Axisymmetric: (1) Two figures symmetrical about a line are congruent; 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 straight line connecting the corresponding points of two graphs is vertically bisected by the same straight line, then 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.

40. Central symmetry: (1) Two figures with central symmetry are congruent; (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, similarity: (1) If two graphs are not only similar, but also each group of straight lines corresponding to vertices pass through the same point, then such two graphs are called similarity graphs, and this point is called similarity center, and the similarity ratio at this time is also called similarity ratio; (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.

Theorem, axiom and definition of triangle in junior middle school

1. Axioms and theorems in triangles: the properties of the outer angles of triangles (1);

(1) An outer angle of a triangle is equal to the sum of two inner 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) Theorem of the sum of triangle internal angles: the sum of triangle internal angles is equal to 180.

(3) The relationship among the three sides of a triangle: the sum of the two sides is greater than the third side, and the difference between the two sides is less than the third side.

(4) Triangle midline theorem: the midline of a triangle is parallel to the third side and equal to half of the third side. 2. Relevant axioms and theorems in polygons:

(1) theorem of interior angles of polygons: the sum of interior angles of n polygons is equal to (n-2) × 180. (2) Theorem of the sum of external angles of polygons: the sum of external angles of any polygon is 360.

(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 symmetry axis of an axisymmetric figure is the median vertical line of the line segment connected by any pair of corresponding points. 4. Axioms and theorems in an isosceles 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) The theorem of "three lines in one" of isosceles triangle: the bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of isosceles triangle coincide, 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) A triangle with three equal angles is an equilateral triangle.

(6) An isosceles triangle with an angle of 60 is an equilateral triangle. 5. Axioms and theorems about right triangle: (1) The two acute angles of 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 triangle, if an acute angle is equal to 30, then the right side it faces is equal to half of the hypotenuse. Judgment of intransitive verb similar triangles;

(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. (4) The triangle formed by the intersection of a straight line parallel to one side of the triangle and the other two sides is similar to the original triangle. Seven. The edges and angles of congruent polygons are equal respectively. Eight. 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, they are congruent. (A.A.S.) (3) Two triangles are identical if their two angles are equal to the side they sandwich.

(5) If the hypotenuse and a right-angled side of two right-angled triangles are equal respectively, the two right-angled triangles are congruent. (H.L.) IX。 The concept of angle.

The concept of junior middle school angle is: a graph composed of two rays with a common endpoint is called an angle;