Two triangles with two sides and their included angles equal.
23
Corner axiom (
ASA) Two triangles with two corners, and their clamping edges are equal.
24
reason
There are two angles, and the opposite side of one angle corresponds to the coincidence of two triangles.
25
Edge axiom
Two triangles with three corresponding equilateral sides are congruent.
26
Axiom of hypotenuse and right angle (HL)
Two right-angled triangles with a hypotenuse and a right-angled side are congruent.
27
Theorem 1
A point on the bisector of an angle is equal to the distance on both sides of the angle.
28
Theorem 2
A point on the bisector of an angle that is equidistant from both sides of the angle.
29
The bisector of an angle is the set of all points with equal distance to both sides of the angle.
30
Property theorem of isosceles triangle
The two base angles of an isosceles triangle are equal.
(i.e. equilateral and equiangular)
3 1
Inference 1
The bisector of the vertices of an isosceles triangle bisects and is perpendicular to the bottom.
32
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.
33
Inference 3
All angles of an equilateral triangle are equal, and each angle is equal to 60.
34
Judgement theorem of isosceles triangle
If the two angles of a triangle are equal, then the opposite sides of the two angles are also equal (equilateral).
35
Inference 1
A triangle with three equal angles is an equilateral triangle.
36
reason
2
An isosceles triangle with an angle equal to 60 is an equilateral triangle.
37
In a right triangle, if an acute angle is equal to 30, the right side it faces is equal to half of the hypotenuse.
38
The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
39
theorem
The point on the vertical line of a line segment is equal to the distance between the two endpoints of the line segment.
40
Inverse principle
On the perpendicular bisector of a line segment, a point equidistant from the two endpoints of the line segment.
4 1
The middle vertical line of a line segment can be regarded as a set of all points with the same distance at both ends of the line segment.
Forty two.
Theorem 1
The congruence of two figures symmetric about a straight line.
43
theorem
2
If two figures are symmetrical about a straight line, then the symmetry axis is the middle vertical line connecting the corresponding points.
Theorem 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.
45 inverse theorem
If the straight 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.
46 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, A 2+B 2 = C 2.
47 Inverse Theorem of Pythagorean Theorem
If the three sides of a triangle are A, B and C, then A 2+B 2 = C 2.
Then this triangle is a right triangle.
Theorem 48
The sum of the internal angles of a quadrilateral is equal to 360 degrees.
The sum of the external angles of the quadrilateral is equal to 360.
Theorem of sum of internal angles of 50 polygons
The sum of the internal angles of an N-polygon is equal to (n-2) × 180.
5 1 reasoning
The sum of the outer angles of any polygon is equal to 360 degrees.
52 parallelogram property theorem 1
The diagonals of parallelogram are equal.
53 parallelogram property theorem 2
The opposite sides of a parallelogram are equal.
54 inference
The parallel segments sandwiched between two parallel lines are equal.
55 parallelogram property theorem 3
Diagonal lines of parallelograms are equally divided.
56 parallelogram judgment theorem 1
Two sets of diagonally equal quadrilaterals are parallelograms.
57 parallelogram decision theorem 2
Two sets of quadrilaterals with equal opposite sides are parallelograms.
58 parallelogram decision theorem 3
Quadrilaterals whose diagonals bisect each other are parallelograms.
59 parallelogram decision theorem 4
A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.
60 rectangle property theorem 1
All four corners of a rectangle are right angles.
6 1 rectangle property theorem 2
Diagonal lines of rectangles are equal.
62 rectangle judgment theorem 1
A quadrilateral with three right angles is a rectangle.
63 Rectangular Decision Theorem 2
A parallelogram with equal diagonal lines is a rectangle.
64 diamond property theorem 1
All four sides of a diamond are equal.
65 diamond property theorem 2
Diagonal lines of the diamond are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.
66 Diamond area = half of diagonal product, that is, S=(a×b)÷2.
67 diamond decision theorem 1
A quadrilateral with four equilateral sides is a diamond.
68 Diamond Decision Theorem 2
Parallelograms with diagonal lines perpendicular to each other are diamonds.
69 square property theorem 1
All four corners of a square are right angles and all four sides are equal.
Theorem of 70 Square Properties 2 Two diagonal lines of a square are equal and bisected vertically, and each diagonal line bisects a set of diagonal lines.
Theorem 7 1 1
On the congruence of two graphs with central symmetry.
Theorem 2
For two graphs with central symmetry, the straight lines connecting the symmetrical points pass through and are equally divided by the symmetrical center.
73 inverse theorem
If the lines connecting the corresponding points of two graphs pass through a certain point and are bounded by this point.
If the point is split in two, then the two graphs are symmetrical about the point.
74 isosceles trapezoid property theorem
The two angles of the isosceles trapezoid with the same base are equal.
The two diagonals of an isosceles trapezoid are equal.
76 isosceles trapezoid judgment theorem
A trapezoid with two equal angles on the same base is an isosceles trapezoid.
A trapezoid with equal diagonal lines is an isosceles trapezoid.
The bisection theorem of 78 parallel lines
If a set of parallel lines is a line segment on a straight line.
Equal, then the line segments cut on other straight lines are also equal.
79
Inference 1
A straight line passing through the midpoint of one waist of the trapezoid and parallel to the bottom will bisect the other waist.
80
Inference 2
A straight line passing through the midpoint of one side of a triangle and parallel to the other side will be equally divided.
Trilaterality
8 1
Triangle midline theorem
The center line of the triangle is parallel to and equal to the third side.
Half of
82
Trapezoidal midline theorem
The center line of the trapezoid is parallel to the two bottom sides and is equal to the sum of the two bottom sides.
one half
L=(a+b)÷2
S = length × height
83
Basic properties of (1) ratio
If a:b=c:d, then ad=bc.
If ad=bc, then a: b = c: d.
84
(2) Comprehensive performance
If a/b = c/d, then (a b)/b = (c d)/d.
Eighty-five
(3) Equidistant property
If a/b = c/d = … = m/n (b+d+…+n ≠ 0), then
(a+c+…+m)/(b+d+…+n)=a/b
86
Proportional theorem of parallel line segment
Three parallel lines cut two straight lines, corresponding.
The line segments are proportional.
87
reason
A straight line parallel to one side of a triangle cuts the other two sides (or extension lines on both sides), and the corresponding line segments are proportional.
88
theorem
If the corresponding line segments obtained by cutting two sides of a triangle (or the extension lines of two sides) are proportional, then this line is parallel to the third side of the triangle.
Eighty-nine
A straight line parallel to one side of a triangle and intersecting with the other two sides. The three sides of the triangle are proportional to the original three sides.
90
theorem
A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines of both sides) to form a triangle similar to the original triangle.
9 1
The Judgment Theorem of similar triangles 1
Two angles are equal and two triangles are similar (ASA)
92
Two right triangles divided by the height on the hypotenuse are similar to the original triangle.
93
Decision theorem 2
The two sides are proportional and the included angle is equal, and the two triangles are similar (SAS).
94
Decision theorem 3
Three sides are proportional and two triangles are similar (SSS)
95
theorem
If the hypotenuse of a right triangle and one right-angled side and another right-angled side.
The hypotenuse of an angle is proportional to a right-angled side, so two right-angled triangles are similar.
96
Property theorem 1
Similar triangles has a high proportion, and the proportion corresponding to the center line is equal to the corresponding angle.
The ratio of dividing lines is equal to the similarity ratio.
97
Property theorem 2
The ratio of similar triangles perimeter is equal to the similarity ratio.
98
Property theorem 3
The ratio of similar triangles area is equal to the square of similarity ratio.
99
The sine value of any acute angle is equal to the cosine value of other angles, the cosine value of any acute angle, etc.
Sine value of other angles
100 The tangent of any acute angle is equal to the cotangent of other angles, the cotangent of any acute angle, etc.
Tangent value of its complementary angle
10 1 A circle is a set of points whose distance from a fixed point is equal to a fixed length.
102 The interior of a circle can be regarded as a collection of points whose center distance is less than the radius.
The outer circle of 103 circle can be regarded as a collection of points whose center distance is greater than the radius.
104 The radius of the same circle or equal circle is the same.
105 The distance from the fixed point is equal to the trajectory of the fixed point, with the fixed point as the center, and the fixed length is half.
Diameter circle
106 and it is known that the locus of the point with the same distance between the two endpoints of the line segment is perpendicular to the line segment.
bisector
The locus from 107 to a point with equal distance on both sides of a known angle is the bisector of this angle.
The trajectory from 108 to the point with the same distance from two parallel lines is parallel to these two parallel lines with a distance of.
A straight line of equality
Theorem 109
Three points that are not on the same straight line determine a circle.
1 10 vertical diameter theorem
The diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.
1 1 1 inference 1
(1) bisects the diameter (not the diameter) of the chord perpendicular to the chord and bisects the two arcs opposite to the chord.
(2) The perpendicular line of the chord passes through the center of the circle and bisects the two arcs opposite to the chord.
③ bisect the diameter of an arc opposite to the chord, bisect the chord vertically, and bisect another arc opposite to the chord.
1 12 Inference 2
Two parallel chords of a circle have equal arcs.
1 13 circle is a centrosymmetric figure with the center of the circle as the symmetry center.
Theorem 1 14
In the same circle or in the same circle, the arcs with equal central angles are equal, and so are the chords.
Equal, the chord center distance of the opposite chord is equal.
1 15 inference
In the same circle or equal circle, if two central angles, two arcs, two chords or two
If one set of quantities in the chord-to-chord distance is equal, then the other sets of quantities corresponding to it are also equal.
Theorem 1 16
An arc subtends a circumferential angle equal to half the central angle it subtends.
1 17 inference 1
The circumferential angles of the same arc or equal arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.
1 18 inference 2
The circumference angle (or diameter) of a semicircle is a right angle; 90 degree circle angle
The chord on the right is the diameter.
1 19 inference 3
If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.
Theorem 120
The diagonals of the inscribed quadrilateral of a circle are complementary, and any external angle is equal to it.
Internal diagonal of
12 1① the straight line l intersects with ⊙ o.
d