Part I: Points, Lines and Angles
I. Production line
1, line 2, line 3, line segment
Second, the angle
1, there are two definitions of an angle: one is that a graph composed of two rays with a common endpoint is called an angle.
The other is a graph formed by light rotating from one position to another around the endpoint.
2. The bisector of the angle
3. Measurement of angle: To measure angle, you can use "degree" as the unit of measurement. Divide a circle into 360 equal parts, and each equal part is called an angle of one degree. 1 degree =60 points; 1 min =60 seconds.
4. Classification of angles: (1) acute angle (2) right angle (3) obtuse angle (4) right angle (5) rounded corner.
5. Relevant angles:
(1) Vertex (2) is complementary angle (3) is complementary angle.
6. Adjacent complementary angles: there is a common vertex, a common edge, and the other two edges are opposite to the extension line. The two angles are mutually adjacent complementary angles.
Note: Complementarity and complementarity refer to the quantitative relationship between two angles, which has nothing to do with the position of the two angles, while adjacency requires the two angles to have a special position relationship.
7, the nature of the angle
(1) Equal vertex angle (2) Equal angle or equal angle (3) Equal angle or equal angle.
Third, the intersection line
1, diagonal 2, two perpendicular straight lines 3, vertical line, vertical foot
4. Nature of vertical line
(l) There is one and only one straight line perpendicular to the known straight line.
(2) The vertical segment is the shortest.
Fourth, distance.
1, the distance between two points
2. The length from a point outside a straight line to the vertical section of the straight line is called the distance from the point to the straight line.
3. Distance between two parallel lines: Two parallel lines are parallel, and draw a vertical line from any point on one line to another. The length of a vertical line segment is called the distance between two parallel lines.
Five, parallel lines
1. Definition: In the same plane, two lines that do not intersect are called parallel lines.
Note: It can also be said that two rays or two line segments are parallel, in fact, it means that their straight lines are parallel.
2. Determination of parallel lines:
(1) At the same angle, two straight lines are parallel.
(2) The internal dislocation angles are equal and the two straight lines are parallel.
(3) Parallel to two straight lines complementary to the inner corner of the side surface.
3. The nature 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.
Note: Prove that two straight lines are parallel, and use judgment axiom (or theorem). When two straight lines are parallel under known conditions, the property theorem is applied.
4. If both sides of an angle are parallel to both sides of another angle, then these two angles are _ _ _ _ _ _ _ _ _ _ _.
5. If both sides of an angle are perpendicular to both sides of another angle, then these two angles are _ _ _ _ _ _ _ _ _.
Part II: Triangle
Knowledge points:
First of all, some concepts about triangles.
1, the bisector of the triangle.
The bisector of a triangle is a line segment (the distance from the vertex to the bisector of the inner corner and the intersection of the opposite sides).
The bisectors of the three angles intersect at a point (the intersection point is inside the triangle and is the center of the inscribed circle of the triangle, which is called the heart).
2. The center line of the triangle
The center line of a triangle is also a line segment (the distance from the vertex to the midpoint of the opposite side).
The three median lines intersect at a point (the intersection point is inside the triangle and is the geometric center of the triangle, called the center).
3. The height of the triangle
The height of a triangle is also a line segment (the distance from the vertex to the opposite side).
Note: The midline and bisector of the triangle are both within the triangle.
As shown in Figure 2-l, AD, BE and CF are all angular bisectors of ABC, which are all within △ABC.
As shown in Figure 2-2, AD, BE and CF are all the center lines of △ABC, and they are all within △ABC.
Figure 2-3 shows that the high line is not necessarily within △ABC.
Figure 2-3-( 1) Figure 2-3-(2) Figure 2-3-(3)
Figure 2-3-( 1), the three high lines are all within △ ABC,
Figure 2-3-(2), the medium and high line CD is within △ABC, and the high lines AC and BC are the sides of a triangle;
In Figure 2-3-(3), the medium-high line BE is within △ABC, and the high lines AD and CF are outside △ABC.
Second, the relationship between the three sides of a triangle
The three sides of a triangle are not equal, which is called an equilateral triangle; An isosceles triangle with two equal sides; A triangle with three equilateral sides is called an equilateral triangle.
In an isosceles triangle, two equal sides are called waist, the other side is called bottom, the angle between waist and bottom is called bottom angle, and the angle between two waist is called back neck angle.
Triangular classification
Classification according to the equal relationship between adjacent edges:
Represented by a set, as shown in Figure 2-4.
It is inferred that the difference between the two sides of a triangle is smaller than the third side.
Three line segments that do not conform to the theorem cannot form three sides of a triangle.
For example, the lengths of the three lines are 5,6, 1 respectively, because 5+6.
Third, the sum of the internal angles of the triangle.
Theorem The sum of three internal angles of a triangle is equal to 180.
According to the theorem, only one of the three internal angles of a triangle can be right angle or obtuse angle.
Inference 1: The two acute angles of a right triangle are complementary.
Triangles are classified by angle:
Represented by a set, as shown in the figure.
The angle formed by one side of a triangle and the extension line of the other side is called the outer angle of the triangle.
Inference 2: One external angle of a triangle is equal to the sum of two non-adjacent internal angles.
Inference 3: An outer angle of a triangle is larger than any inner angle that is not adjacent to it.
For example, in figs. 2-6
∠ 1 >∠3; ∠ 1=∠3+∠4; ∠5 >∠3+∠8; ∠5=∠3+∠7+∠8;
∠2 >∠8; ∠2=∠7+∠8; ∠4 >∠9; ∠4=∠9+∠ 10 and so on.
Fourth, congruent triangles.
Two figures that can completely overlap are called conformal.
When two congruent triangles overlap, the overlapping vertices are called corresponding vertices, the overlapping edges are called corresponding edges, and the overlapping angles are called corresponding angles.
The corresponding sides of congruent triangles are equal; Congruent triangles's corresponding angles are equal.
Verb (abbreviation of verb) congruent triangles's judgment
1, edge axiom: "SAS"
Note: it must be the included angle between two sides, not the edge and angle.
2. Angular axioms: ASA 3, AAS 4, SSS
3. Determination of congruence of right-angled triangle: hypotenuse, right-angled edge or HL.
The important property of triangle: the stability of triangle.
Six, the bisector of the angle
Theorem 1. A point on the bisector of an angle is equal to the distance on both sides of the angle.
Theorem 2. The point where two sides of an angle are equidistant is on the bisector of this angle.
It can be proved that there is a point in the triangle, and its distance to the three sides of the triangle is equal. This point is the intersection of three bisectors of a triangle.
Seven, the determination of isosceles triangle
Theorem: If a triangle has two angles, both sides of the two angles are equal. (abbreviated as "equiangular equivalent motion").
Inference 1: A triangle with three equal angles is an equilateral triangle.
Inference 2: An isosceles triangle with an angle equal to 60 is an equilateral triangle.
Inference 3: In a right triangle, if an acute angle is equal to 3o, then the right-angled side it faces is equal to half of the hypotenuse.
Eight, Pythagorean theorem
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;
Inverse Pythagorean Theorem: If three sides of a triangle have the following relationship:
So this triangle is a right triangle.