1. There is only one straight line after two o'clock.

(Jane: Two points are in a straight line)

2. The shortest line segment between two points.

3. The complementary angles of the same angle or equal angle are equal.

The same angle or the complementary angle of the same angle is equal.

4. There is one and only one straight line perpendicular to the known straight line.

5. The vertical line segment is the shortest among all the line segments connecting a point outside the line with points on the line.

Judgment of parallel lines

1. The parallel axiom passes through a point outside a straight line, and one and only one straight line is parallel to this straight line.

2. If two straight lines are parallel to the third straight line, then the two straight lines are also parallel to each other (Jane: two straight lines parallel to the same straight line are parallel).

3. The same angle is equal and two straight lines are parallel.

4. The internal dislocation angles are equal and the two straight lines are parallel.

5. The internal angles on the same side are complementary and the two straight lines are parallel.

Properties of parallel lines

1. Two straight lines are parallel with the same angle.

2. The two straight lines are parallel and the internal dislocation angles are equal.

3. Two straight lines are parallel and complementary.

The relationship between the three sides of a triangle

1. The sum of two sides of the triangle is greater than the third side, and the difference between the two sides of the triangle is less than the third side.

The relationship between triangle angles

1. The sum of the internal angles of the triangle is equal to 180.

2. The two acute angles of a right triangle are complementary.

3. The outer angle of a triangle is equal to the sum of two non-adjacent inner angles.

The outer angle of a triangle is greater than any inner angle that is not adjacent to it.

Congruent triangles's Nature and Judgment

1. congruent triangles has equal sides and angles.

2. The Axiom of Angle and Edge (SAS) has the consistency of two equilateral triangles.

3. Angle and Angle Axiom (ASA) has congruence of two triangles, which have two angles and their sides correspond to each other.

4. Reasoning (AAS) has two angles, and the opposite side of one angle corresponds to the coincidence of two triangles.

5. The edge axiom (SSS) has two triangles and three corresponding equilateral sides.

6. Axiom of hypotenuse and right angle (HL) Two right angle triangles with hypotenuse and right angle are congruent.

The nature and judgment of angular bisector

Property: The distance between points on the bisector of an angle is equal to both sides of the angle.

Judgment: The points with equal distance on both sides of an angle are on the bisector of this angle.

Properties of isosceles triangle

1. The property theorem of isosceles triangle. The two base angles of an isosceles triangle are equal (that is, equilateral and equilateral).

2. It is inferred that the bisector of the vertex of 1 isosceles triangle bisects the base and is perpendicular to the base.

3. 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. Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60.

Isosceles triangle judgment

1 Decision Theorem of isosceles triangle If a triangle has two equal angles, then the opposite sides of the two angles are also equal (equal angles and equal sides).

2. A triangle with three equal angles is an equilateral triangle.

An isosceles triangle with an angle equal to 60 is an equilateral triangle.

Nature and judgment of vertical line in line segment

1. Theorem: The distance between a point on the vertical line of a line segment and its two endpoints is equal.

2. Inverse theorem: The point where the distance between two endpoints of a line segment is equal is on the middle vertical line of this line segment.

3. The middle vertical line of a line segment can be regarded as the set of all points with the same distance at both ends of the line segment.

Axial symmetry, central symmetry, translation and rotation

1. The congruence of two graphs symmetric about a straight line.

2. If two figures are symmetrical about a straight line, then the symmetry axis is the middle vertical line connecting the corresponding points.

3. Two figures are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is 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.

5. Two centrally symmetric graphs are congruent.

For two graphs with symmetrical centers, the straight line connecting the symmetrical points passes through the symmetrical center and is equally divided by the symmetrical center.

6. 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 centrosymmetric about the point.

7. The graphics before and after translation or rotation remain unchanged. Central symmetry is a special form of rotation.

Pythagorean Theorem The sum of squares of two right angles A and B of a right triangle is equal to the square of hypotenuse C, that is, a2+b2=c2.

Inverse Theorem of Pythagorean Theorem If the lengths of three sides of triangle A, B and C are related to a2+b2=c2, then this triangle is a right triangle. (1) in a right triangle, if an acute angle is equal to 30, the right side it faces is equal to half of the hypotenuse.

② The median line on the hypotenuse of a right triangle is equal to half of the hypotenuse.

The sum of the inner and outer angles of N-sided and quadrilateral.

The sum of the internal angles of the 1. quadrilateral is equal to 360 degrees.

2. The sum of the external angles of the quadrilateral is equal to 360.

3. Theorem The sum of the internal angles of a polygon and an n-sided polygon is equal to (n-2) 180.

4. Infer that the sum of the external angles of any polygon is equal to 360.

Parallelogram property

1. The diagonals of the parallelogram are equal.

2. The opposite sides of the parallelogram are equal.

3. The parallel segments sandwiched between two parallel lines are equal.

4. The diagonal bisection of parallelogram.

Parallelogram judgment

1. Two groups of parallelograms with parallel opposite sides are parallelograms.

2. Two sets of quadrangles with equal diagonals are parallelograms. 3. Two groups of quadrilaterals with equal opposite sides are parallelograms.

4. Quadrilaterals whose diagonals are bisected are parallelograms.

5. A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.

Rectangular attribute

1. All four corners of a rectangle are right angles.

2. The diagonals of rectangles are equal.

Rectangular judgment

1. A parallelogram with right angles is a rectangle.

A quadrilateral with three right angles is a rectangle.

3. A parallelogram with equal diagonals is a rectangle.

Diamond attribute

1, all four sides of the diamond are equal.

2. The diagonals of the diamond are perpendicular to each other, and each diagonal bisects a set of diagonals.

3, diamond area = half of the diagonal product, namely

Diamond decision

1. A set of parallelograms with equal adjacent sides is a diamond.

2. A quadrilateral with four equilateral sides is a diamond.

3. Parallelograms with diagonal lines perpendicular to each other are rhombic.

Square attribute

1. All four corners of a square are right angles and all four sides are equal.

2. The two diagonals of a square are equal and bisected vertically, and each diagonal bisects a set of diagonals.

A fair judgment

1. A quadrilateral with four right angles and four equal sides is a square.

2. The quadrilateral whose diagonal lines are vertically bisected and equal is a square.

Isosceles trapezoid property

1. The two angles of the isosceles trapezoid with the same base are equal.

2. The two diagonals of the isosceles trapezoid are equal.

Isosceles trapezoid judgment

1. A trapezoid with two equal angles on the same base is an isosceles trapezoid.

2. A trapezoid with equal diagonal lines is an isosceles trapezoid.

A straight line passing through the midpoint of the trapezoid waist and parallel to the bottom will bisect the other waist.

A straight line passing through the midpoint of one side of a triangle and parallel to the other side will bisect the third side.

The midline theorem of triangle: the midline of triangle is parallel to the third side and equal to half of it.