Knowledge points-intersection lines and parallel lines
1. Theorem and Properties
The nature of antipodal angle: antipodal angle is equal.
2. The nature of the vertical line:
Property 1: There is one and only one straight line perpendicular to the known straight line.
Property 2: Of all the line segments connecting a point outside the straight line and a point on the straight line, the vertical line segment is the shortest.
3. Parallelism axiom: One and only one straight line is parallel to the known straight line through a point outside the straight line.
Inference of the axiom of parallelism: If two straight lines are parallel to the third straight line, then the two straight lines are also parallel to each other.
4. The essence of parallel lines:
Property 1: Two straight lines are parallel and equal to the complementary angle.
Property 2: Two straight lines are parallel and the internal dislocation angles are equal.
Property 3: Two straight lines are parallel and complementary.
5. Determination of parallel lines:
Judgment 1: congruent angles are equal and two straight lines are parallel.
Decision 2: The internal dislocation angles are equal and the two straight lines are parallel.
Judgment 3: The internal angles on the same side are equal and the two straight lines are parallel.
Knowledge point two triangle
First, the related concepts of triangle
1. The concept of triangle is called triangle, which consists of three line segments that are not on the same straight line.
Key points: ① Three line segments; ② Not in a straight line; ③ End-to-end connection.
2. Three important segments in the triangle
(1) Angle bisector of a triangle: the bisector of an 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 angle bisector of the triangle.
(2) The center line of a triangle: In a triangle, the line segment connecting the vertex and the midpoint of its opposite side is called the center line of the triangle.
(3) Height line of triangle: Draw a vertical line from a vertex of the triangle to its opposite side, and the boundary between the vertex and the vertical foot is called the height line of the triangle.
Second, the triangle trilateral relation theorem
① The sum of the two sides of a triangle is greater than the third side, so the inequality of the three-side lengths A, B and C that satisfy △ABC at the same time is: A+B >; c，b+ c & gt； a，c+a & gt； b.
(2) The difference between two sides of a triangle is smaller than the third side, so the inequalities that satisfy the three-side lengths A, B and C of △ABC at the same time are: a & gtb-c, b & gta-c, C>B-A..
Note: To judge whether these three line segments can form a triangle, we only need to see whether the sum of the lengths of the two shorter line segments is greater than the third line segment.
Thirdly, the stability of triangle.
When the three sides of a triangle are determined, its shape and size are also determined. This property of triangle is called the stability of triangle. For example, the crane bracket adopts a triangular structure.
Fourth, the inner angle of the triangle.
Conclusion 1: The sum of interior angles of triangle is 180, that is, in △ABC, ∠ A+∠ B+∠ C = 180.
Conclusion 2: In a right triangle, the two acute angles are complementary.
Note: ① In a triangle, if two internal angles are known, the third internal angle can be found.
For example, in △ABC, ∠ C = 180-(∠ A+∠ B).
(2) in a triangle, the ratio of the sum of three internal angles or the relationship between them is known, and each internal angle is found.
For example, in △ABC, it is known that ∠A: ∠B: ∠C = 2: 3: 4, and the number of times ∠A, ∠B and ∠C are found.
Verb (abbreviation for verb) The outer corner of a triangle.
1. Meaning: 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.
2. Nature:
An 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.
③ One outer corner of a triangle is complementary to its adjacent inner corner.
Six, polygon
(1) the diagonal of the polygon is diagonal; ② The sum of the internal angles of the N-polygon is (n-2) ×180; ③ The sum of the external angles of the polygon is 360.
Three congruent triangles of knowledge points
Congruent triangle
1, "congruence" Understanding congruence graphs must meet the following requirements: (1) Graphs with the same shape; (2) Graphics of equal size;
That is, two figures that can completely overlap are called conformal. Similarly, we say that two triangles can completely overlap the congruent triangles.
2. The nature of congruent triangles
(1) The corresponding edges of congruent triangles are equal; (2) The angles corresponding to congruent triangles are equal;
Congruent triangles's judgment method.
(1) Three sides correspond to the congruence of two triangles. (SSS)
(2) The two corners and their clamping edges are congruent with each other. (ASA)
(3) The opposite sides of two angles and one angle correspond to the congruences of two equal triangles. (AAS)
(4) Two triangles with equal angles between two sides. (SAS)
(5) The hypotenuse and the right-angled side correspond to the congruence of two right-angled triangles. (HL)
4. The nature and judgment of angular bisector
Property: The point on the bisector of the angle is equal to the distance on both sides of the angle.
Judgment: The points with equal distance to both sides of an angle are on the bisector of this angle.
Second, axisymmetric graphics.
(A) the basic definition
1. Axisymmetric graph
If a figure is folded along a straight line, the parts on both sides of the straight line can overlap each other. This figure is called an axisymmetric figure, and this straight line is called an axis of symmetry. The overlapping points after folding are the corresponding points, which are called symmetrical points.
2. perpendicular bisector of the line segment
A straight line that passes through the midpoint of a line segment and is perpendicular to the line segment is called the midline of the line segment.
3. Axisymmetric transformation
The axisymmetric figure obtained from plane figure is called axisymmetric transformation.
4. 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 sides is called the vertex angle, and the angle between the lower side and the waist is called the bottom angle.
5. equilateral triangle
A triangle with three equilateral sides is called an equilateral triangle.
(2) Nature
1. If two graphs are symmetrical about a straight line, then the symmetry axis is the median vertical line of the line segment connected by any pair of corresponding points, or the symmetry axis of an axisymmetric graph is the median vertical line of the line segment connected by any pair of corresponding points.
2. Divide the nature of money vertically with line segments.
The point on the vertical line in a line segment is equal to the distance between the two endpoints of the line segment.
3.( 1) Point P(x, y) The coordinate of the point that is symmetrical about X axis is P ′ (x, -y).
(2) The coordinate of the point P(x, y) about the y axis symmetry is p "(-x, y).
4. The nature of isosceles triangle
(1) The two base angles of an isosceles triangle are equal.
(2) 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.
(3) An isosceles triangle is an axisymmetric figure, and the straight line where the median line on the bottom (bisector of the height of the top corner and the bottom) is located is its axis of symmetry.
(4) The height and midline of the two waists of an isosceles triangle are equal, and the bisectors of the two bottom angles are equal.
(5) The included angle between the height of the isosceles triangle and the waist bottom is half of the vertex angle.
(6) The bisector of the outer angle of the vertex of the isosceles triangle is parallel to the base of the triangle.
5. Properties of equilateral triangles
(1) All three internal angles of an equilateral triangle are equal, and each angle is equal to 60.
(2) An equilateral triangle is an axisymmetric figure, and * * * has three axes of symmetry.
(3) The midline and height of each side of an equilateral triangle coincide with the bisector of the inner angle of that side.
(3) Relevant judgments
1. The points that are equidistant from the two endpoints of a line segment are on the perpendicular line of this line segment.
2. If the two angles of a triangle are equal, the opposite sides of the two angles are also equal (abbreviated as "equilateral").
A triangle with three equal angles is an equilateral triangle.
An isosceles triangle with an angle of 60 is an equilateral triangle.
Knowledge points of the Four Pythagorean Theorem
1, Pythagorean theorem definition: If the lengths of two right angles of a right triangle are A and B, and the length of the hypotenuse is C, then
A2+B2 = C2。 That is to say, the sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse.
Hook: The shorter right side of a right triangle.
Side: The longer right side of a right triangle.
Chord: hypotenuse
Inverse theorem of Pythagorean theorem: If three sides of a triangle have the following relationship: A2+B2 = C2, then this triangle is a right triangle.
2. Pythagorean numbers: Three positive integers satisfying A2+B2 = C2 are called Pythagorean numbers (Note: If A, B and C are Pythagorean numbers, then ka, kb and kc are also Pythagorean arrays. )
* Attachment: Common Pythagorean numbers: 3, 4, 5; 6,8, 10; 9, 12, 15; 5, 12, 13
3. Judging a right triangle: If three sides of a triangle meet a2+b2=c2, then the triangle is a right triangle. (Classic right triangle: hook three, string four, string five)