Summary of knowledge points in the first volume of eighth grade mathematics published by People's Education Press
Chapter 11 Triangle
I. Knowledge framework:
Second, the concept of knowledge:
1. triangle: A figure composed of three line segments that are not on the same line and are connected end to end is called a triangle.
2. Trilateral relationship: the sum of any two sides of a triangle is greater than the third side, and the difference between any two sides is less than the third side.
3. Height: Draw a vertical line from the vertex of the triangle to the line where the opposite side is located, and the line segment between the vertex and the vertical foot is called the height of the triangle.
4. midline: in a triangle, the line segment connecting the vertex and its relative midpoint is called the midline of the triangle.
5. Angular bisector: The bisector of the inner angle of a triangle intersects the opposite side of this angle, and the line segment between the intersection of the vertex and this angle is called the angular bisector of the triangle.
6. Stability of triangle: The shape of triangle is fixed, and this property of triangle is called stability of triangle.
7. Polygon: On the plane, a figure composed of some line segments connected end to end is called polygon.
8. Interior Angle of Polygon: The angle formed by two adjacent sides of a polygon is called its interior angle.
9. Exterior angle of polygon: The angle formed by the extension line of one side of polygon and its adjacent side is called the exterior angle of polygon.
10. Diagonal line of polygon: the line segment connecting two nonadjacent vertices of polygon is called diagonal line of polygon.
1 1. Regular polygon: A polygon with equal angles and sides in a plane is called a regular polygon.
12. Plane mosaic: A part of a plane is completely covered by some non-overlapping polygons, which is called polygon coverage plane (plane mosaic). Mosaic condition: When the internal angles of several polygons that are put together around a point add up to form one, they can be put together into a plane figure.
13. Formulas and properties:
⑴ Sum of triangle internal angles: The sum of triangle internal angles is 180.
(2) the nature of the triangle exterior angle:
Property 1: One outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it.
Property 2: The outer angle of a triangle is larger than any inner angle that is not adjacent to it.
⑶ Formula for the sum of polygon internal angles: the sum of polygon internal angles is equal to 180.
(4) Sum of polygon external angles: the sum of polygon external angles is 360.
5] Diagonal number of polygons: ① A diagonal line can be drawn from a vertex of a polygon to divide the polygon into triangles. ② A polygon * * * has a diagonal.
Chapter 12 congruent triangles
I. Knowledge framework:
Second, the concept of knowledge:
1. Basic definition:
(1) Conformity: two figures can completely overlap and are called congruences.
⑵ congruent triangles: Two triangles that can completely coincide are called congruent triangles.
⑶ Corresponding vertices: The mutually coincident vertices in congruent triangles are called corresponding vertices.
⑷ Corresponding edges: The overlapping edges in congruent triangles are called corresponding edges.
5] Correspondence angle: The mutually coincident angles in congruent triangles are called correspondence angles.
2. Basic nature:
⑴ Stability of the triangle: When the lengths of the three sides of the triangle are determined, the shape and size of the triangle are determined. This property is called the stability of triangles.
⑵ The nature of congruent triangles: the corresponding edges of congruent triangles are equal, and the corresponding angles are equal.
3. congruent triangles's judgment theorem;
⑴ Edge edge (): three sides correspond to the congruence of two triangles.
⑵ Corner (): Two triangles with equal included angles are congruent.
(3) corner (): the intersection of two corners and their edges corresponding to two triangles.
(4) Corner edge (): the opposite side of two angles and one of them corresponds to the congruence of two triangles.
5] hypotenuse and right-angled edge (): hypotenuse and a right-angled edge correspond to the congruence of two right-angled triangles.
4. Angle bisector:
(1) Painting:
⑵ Property theorem: A point on the bisector of an angle is equal to the distance on both sides of the angle.
(3) The inverse theorem of the property theorem: the point with equal distance from the inside of the angle to both sides of the angle is on the bisector of the angle.
5. The basic method of proof:
(1) Make clear what is known and verified in the proposition (including implied conditions, such as the angle relationship implied by the edge, angle, bisector, midline, height and isosceles triangle).
⑵ Draw a picture according to the meaning of the question, and use digital symbols to indicate the known and verified.
(3) After analysis, find out the method of proof from the known and write the proof process.
Chapter 13 Axisymmetric
I. Knowledge framework:
Second, the concept of knowledge:
1. Basic concepts:
(1) Axisymmetric graph: If a graph is folded along a straight line, the parts on both sides of the straight line can overlap each other, and this graph is called an axisymmetric graph.
⑵ Two figures are symmetrical: one figure is folded along a straight line. If it can overlap with another graph, then the two graphs are said to be symmetrical about this line.
(3) The midline of the line segment: the straight line passing through the midpoint of the line segment and perpendicular to the line segment is called the midline of the line segment.
⑷ isosceles triangle: A triangle with two equal sides is called an isosceles triangle. Two equal sides are called waist and the other side is called bottom. The angle between the two waists is called the top angle, and the angle between the buttocks and the waist is called the bottom angle.
5. equilateral triangle: A triangle with three equilateral sides is called an equilateral triangle.
2. Basic nature:
The essence of (1) symmetry;
(1) No matter whether an axisymmetric figure or two figures are symmetrical about a straight line, the axis of symmetry is the perpendicular to the line segment connected by any pair of corresponding points.
② Symmetric figures are congruent.
(2) The nature of the vertical line in the line segment:
(1) The point on the vertical line of the line segment is equal to the distance between the two endpoints of the line segment.
(2) The point with equal distance from the two endpoints of a line segment is on the middle vertical line of this line segment.
(3) Coordinate properties of axisymmetrical points.
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(4) the nature of isosceles triangle:
(1) isosceles triangles have equal waists.
② The two base angles of an isosceles triangle are equal (equilateral and equiangular).
③ The bisector of the top corner of the isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide.
④ The isosceles triangle is an axisymmetric figure, and the symmetry axis is the combination of three lines (1).
5] the properties of equilateral triangle:
① All three sides of an equilateral triangle are equal.
② All three internal angles of an equilateral triangle are equal, equal to 60.
③ There are three lines on each side of an equilateral triangle.
④ The equilateral triangle is an axisymmetric figure, and the symmetry axis is the combination of three lines (three lines).
3. Basic judgment:
Determination of (1) isosceles triangle;
A triangle with equal sides is an isosceles triangle.
(2) If the two angles of a triangle are equal, then the opposite sides of the two angles are also equal (equilateral).
(2) Determination of equilateral triangle:
A triangle with three equilateral sides is an equilateral triangle.
(2) A triangle with three equal angles is an equilateral triangle.
③ An isosceles triangle with an angle of 60 is an equilateral triangle.
4. Basic methods:
(1) perpendicular to a known straight line:
(2) The midline of the known line segment:
(3) Symmetry axis: connect two corresponding points and make the middle perpendicular of the connecting line segment.
(4) Make a symmetrical figure of a known figure about a straight line:
5] Make a point on a straight line to make the sum of the distances from this point to two known points on the same side of the straight line the shortest.
Chapter 14 multiplication, division and factorization of algebraic expressions.
I. Knowledge framework:
Chapter 15 Scores
I. Knowledge framework:
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