Chapter 1 Summary of important knowledge points 1 isosceles triangle
Determination and Properties of (1) Triangular Congruence
Judge:
A triangle with two equal sides is congruent. (SSS)
A triangle with two sides and an equal included angle is congruent. (SAS)
Two triangles with equal angles and sides meet. (ASA)
Two triangles with equal angles and equal sides are the same. (AAS)
Two right-angled triangles with equal hypotenuse and right-angled side are congruent. (HL)
Nature:
The corresponding edges of congruent triangles are equal, and the corresponding angles are equal.
(2) Determination, nature and inference of isosceles triangle.
Properties: The two base angles of an isosceles triangle are equal (equilateral and equiangular).
Judgment: A triangle with two equal angles is an isosceles triangle.
Inference: The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of an isosceles triangle coincide (i.e.? Three lines in one? )
(3) The properties and judging theorem of equilateral triangle.
Property theorem: all three angles of an equilateral triangle are equal, and each angle is equal to 60 degrees; All three sides of an equilateral triangle satisfy? Three lines in one? The nature of; An equilateral triangle is an axisymmetric figure with three axes of symmetry.
Decision Theorem: An isosceles triangle with an angle of 60 degrees is an equilateral triangle. Or a triangle with three equal angles is an equilateral triangle.
Chapter 65438 +0 Summary of Important Knowledge Points 2 Right Triangle
(1) Pythagorean Theorem and Its Inverse Theorem
Theorem: The sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse.
Inverse theorem: If the sum of squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle.
(2) Proposition and Inverse Proposition
Proposition includes two parts: known and conclusion; Inverse proposition is to exchange known and conclusion; The correct inverse proposition is the inverse theorem.
Reciprocal proposition: in two propositions, if the conditions and conclusions of one proposition are the conclusions and conditions of the other, then these two propositions are called reciprocal propositions, and one of them is called the inverse proposition of the other proposition.
Reciprocity theorem: If the inverse proposition of a theorem is proved to be true, then it is also a theorem. These two theorems are called reciprocal theorems, and one theorem is called the inverse theorem of the other theorem.
Remarks: A proposition must have an inverse proposition, but a theorem does not necessarily have an inverse theorem.
(3) Theorem for judging congruence of right triangle.
Theorem: The hypotenuse and a right-angled side correspond to the congruence (HL) of two equal right-angled triangles.
(4) Theorem: The two acute angles of a right triangle are complementary.
(5) The properties of each side of a 30-degree right triangle.
Theorem: In a right triangle, if an acute angle is equal to 30 degrees, then the right side it faces is equal to half of the hypotenuse.
Chapter 65438 +0 Summary of Important Knowledge Points The vertical line of 3 segments
The Nature and Judgment of the Vertical Line in (1) Line Segment
Property: The distance between the point on the vertical line of a line segment and the two endpoints of this line segment is equal.
Judgment: The points with the same distance to the two ends of a line segment are on the middle vertical line of this line segment.
(2) The nature of the perpendicular lines on three sides of a triangle.
The perpendicular bisector of three sides of a triangle intersect at a point, and the distance from the point to the three vertices is equal.
(3) How to draw the midline of a line segment with a ruler and a ruler?
Make an arc with the two endpoints A and B of the line segment as the center and half the length greater than AB as the radius, and the two arcs intersect at points M and N; Make a straight line MN, then the straight line MN is the middle vertical line of the line segment AB.
Chapter 65438 +0 Summary of important knowledge points 4- angle bisector
Properties and Judgement Theorem of (1) Angular bisector
Property: the point on the bisector of an angle is equal to the distance on both sides of the angle;
Judgment: The points with equal distance from both sides of the angle are on the bisector of the angle.
(2) The property theorem of the bisector of three angles of a triangle.
Property: The three bisectors of a triangle intersect at a point, and the distance from the point to the three sides is equal.
(3) How to draw the angular bisector with a ruler?
Chapter 65438 +0 Summary of Important Knowledge Points 5 Application of Ruler Drawing
As we all know, the base and the height on the base of an isosceles triangle are isosceles triangles.
Chapter 1 Summary of Important Knowledge Points 6 Reduction to absurdity
When proving, it is assumed that the conclusion of the proposition is not valid, and then the results that contradict the definition, basic facts, existing theorems or known conditions are deduced, thus proving that the conclusion of the proposition must be valid. This method of proof is called reduction to absurdity.