The first day of the second book mathematical triangle knowledge points
I. Objectives and requirements
1. Know the triangle, know the meaning of the triangle, know the sides, internal angles and vertices of the triangle, and express the triangle in symbolic language.
2. Experience the practical activities of measuring the side length of a triangle and understand the unequal relationship among the three sides of the triangle.
3. Know how to judge whether three line segments can form a triangle, and use it to solve related problems.
4. The interior angle theorem of triangle can be deduced from the properties of parallel lines.
5. Some simple practical problems can be solved by applying the triangle interior angle sum theorem.
Second, the main points
Theorem of sum of interior angles of triangle;
In order to understand the concept of triangle, three bars can be expressed in symbolic language.
Third, difficulties.
The reasoning process of triangle interior angle sum theorem;
Identify all triangles without repetition or omission in specific graphics;
Judging whether three line segments can form a triangle by the unequal relationship of three sides of a triangle.
Fourth, the knowledge framework.
Verb (abbreviation of verb) summary of knowledge points and concepts
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. Classification of triangles
3. Trilateral relationship of triangle: the sum of any two sides of triangle is greater than the third side, and the difference between any two sides is less than the third side.
4. 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.
5. midline: in a triangle, the line segment connecting the vertex and the midpoint of its opposite side is called the midline of the triangle.
6. 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.
7. Significance and practice of high line, middle line and angle bisector.
8. Stability of triangle: The shape of triangle is fixed, and this property of triangle is called stability of triangle.
9. Theorem of the sum of interior angles of triangle: the sum of three interior angles of triangle is equal to 180.
It is inferred that the two acute angles of 1 right triangle are complementary;
Inference 2: One outer angle of a triangle is equal to the sum of two non-adjacent inner angles;
Inference 3: One outer angle of a triangle is larger than any inner angle that is not adjacent to it;
The sum of the inner angles of a triangle is half of the sum of the outer angles.
10. External angle of triangle: the included angle between one side of triangle and the extension line of the other side is called the external angle of triangle.
1 1. The Properties of the Exterior Angle of Triangle
(1) Vertex is the vertex of a triangle, one side is one side of the triangle, and the other side is the extension line of one side of the triangle;
(2) An outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it;
(3) The outer angle of a triangle is greater than any inner angle that is not adjacent to it;
(4) The sum of the external angles of the triangle is 360.
Knowledge points of mathematics in the first semester
Intersecting line and parallel line
1. In the same plane, there are two kinds of positional relationships between two straight lines: intersecting and parallel, and verticality is a special case of intersection.
2. On the same plane, two disjoint straight lines are called parallel lines. If two straight lines have only one common point, they are said to intersect; If two straight lines have no common point, they are said to be parallel.
3. Among the four angles formed by the intersection of two straight lines, two angles with a common vertex and a common edge are adjacent complementary angles. The nature of adjacent complementary angles: complementary adjacent complementary angles. As shown in figure 1, adjacent to each other and adjacent to each other. += 180 ; += 180 ; += 180 ; += 180 。
4. Among the four corners formed by the intersection of two straight lines, two sides of one corner are opposite extension lines of two sides of the other corner, so the two corners are opposite. The nature of antipodal angle: antipodal angle is equal. As shown in figure 1, and they are opposite to each other. =; =。
5. If one of the angles formed by the intersection of two straight lines is a right angle or 90, the two straight lines are said to be perpendicular to each other, and one of them is called the perpendicular of the other. As shown in Figure 2, when = 90, ⊥.
Nature of 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.
Property 3: As shown in Figure 2, when a ⊥ b = = = 90.
Distance from point to straight line: The length from a point outside a straight line to the vertical section of this straight line is called the distance from point to straight line.
6. The basic characteristics of congruent angle, internal dislocation angle and ipsilateral internal angle:
① On the same side of two straight lines (cut lines) and on the same side of the third straight line (cut lines), such two angles are called isosceles angles. In Figure 3, * * * has a pair of isosceles angles; And is an isosceles angle; And are at the same angle; And are at the same angle; And it's the same angle.
(2) Between two straight lines (secant) and on both sides of the third straight line (secant), such two angles are called inscribed angles. In figure 3; * * has a pair of inner corners; Is the inner corner; And is an inner corner.
(3) Between two straight lines (intersecting lines), both are on the same side of the third straight line (intersecting line), and such two angles are called ipsilateral inner angles. In figure 3; * * has a pair of inner corners on the same side; And are internal angles on the same side; And it is the same inner angle.
7. Parallelism axiom: At a point outside a straight line, only one straight line is parallel to the known 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.
Properties of parallel lines:
Property 1: Two straight lines are parallel and equal to the complementary angle. As shown in figure 4, if a∨b, then =; =; =; =。
Property 2: Two straight lines are parallel and the internal dislocation angles are equal. As shown in figure 4, if a∨b, then =; =。
Property 3: Two straight lines are parallel and complementary. As shown in figure 4, if a∨b,+=180; += 180 。
Property 4: Two lines parallel to the same line are parallel to each other. If a∨b and a∨c, then ∨.
8. Determination of parallel lines:
Judgment 1: congruent angles are equal and two straight lines are parallel. As shown in fig. 5, if = or = or = or =, then a ∑ b.
Decision 2: The internal dislocation angles are equal and the two straight lines are parallel. As shown in figure 5, if = or =, then a ∨ b.
Judgment 3: The internal angles on the same side are complementary and the two straight lines are parallel. As shown in figure 5, if+=180; += 180, then a ∨ b.
Decision 4: Two straight lines parallel to the same straight line are parallel to each other. If a∨b and a∨c, then ∨.
Review knowledge points of seventh grade mathematics
Knowledge point classification-real number
1. Classification by definition: 2. Classification by natural symbols:
Note: 0 is neither positive nor negative.
Knowledge point 2 Related concepts of real numbers
1. Inverse
Algebraic meaning of (1): There are only two numbers with different signs, and we say that one of them is opposite to the other. The antonym of 0 is 0.
(2) Geometric meaning: On both sides of the origin on the number axis, two points with the same distance from the origin represent two opposite numbers, or on the number axis, the points corresponding to two opposite numbers are symmetrical about the origin.
(3) The sum of two opposites is equal to 0.a and B are opposites a+b=0.
2. Absolute value |a|≥0.
3. The reciprocal (1)0 has no reciprocal. (2) Two numbers whose product is 1 are reciprocal. A and b are reciprocal.
4. Square root
(1) If the square of a number is equal to a, it is called the square root of a, a positive number has two square roots, and the two square roots are in opposite directions. 0 has a square root, and the square root itself is 0; Negative numbers have no square root. The square root of a(a≥0) is written as.
(2) The positive square root of a positive number is called the arithmetic square root of a, and the arithmetic square root of a(a≥0) is recorded as.
5. Cubic root
If x3=a, then x is called the cube root of a, and positive numbers have positive cube roots; Negative numbers have negative cubic roots; The cube root of zero is zero.
Knowledge point 3 Real number and axis
Definition of number axis: the straight line defining the origin, positive direction and unit length is called number axis, and the three elements of number axis are indispensable.
Comparison of real numbers of knowledge point four
1. For any two points on the number axis, the point on the right represents a larger number.
2. Positive numbers are all greater than 0, negative numbers are all less than 0, and two positive numbers, the greater the absolute value, the greater the positive number; Two negative numbers; The absolute value is large but small.
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