1, parallel lines
Properties of parallel lines: If two lines are parallel, they are equal in congruence angle, internal dislocation angle and internal angles on the same side are complementary.
Determination of parallel lines: If the isosceles angles are equal, the internal dislocation angles are equal, and the internal angles on the same side are complementary, then two straight lines are parallel.
Distance between parallel lines: the distance between parallel lines is equal everywhere (parallel lines are equal)
Nature of parallel lines: If two lines are parallel, the same position angles are equal, the inner angles are equal, and the inner angles on the same side are complementary.
Judgment of parallel lines: the same angle is equal, the inner angles are equal, and the inner angles on the same side are complementary, so the two lines are parallel.
Distance between parallel lines: the distance between parallel lines is equal everywhere (parallel lines are equal in segment).
Regarding parallel lines, we should pay attention to the use of parallel lines in parallelogram, rectangle and square in our common geometric figures. Students should clearly know that there are a large number of triangles with the same base height that can be used. After finding the same base height, the problem can be easily solved by area equivalent substitution.
Step 2: Triangle
The sum of the internal angles of a triangle is equal to 180 (the sum of the internal angles of a polygon is (n-2) × 180).
The sum of the outer angles of a triangle is equal to 360 (the sum of the outer angles of a polygon is equal to 360).
The external angle of a triangle is equal to the sum of two external angles that are not adjacent to it (a quadrilateral satisfying a four-point circle, the external angle of one angle is equal to the internal angle of the diagonal of the angle. )
The sum of any two sides of a triangle is greater than the third side, and the difference between any two sides is equal to the third side (converted into inequality, we can get such a simple inequality group, a+b >; c，a+c & gt； b，b+ c & gt； A, a, b, c, these are the three sides of a triangle)
The sum of the interior angles of a triangle is equal to 180 (the sum of the interior angles of a polygon: (n-2) × 180).
The sum of the outer angles of a triangle is 360 degrees (the sum of the outer angles of a polygon is 360 degrees).
The outer angle of a triangle is equal to the sum of two non-adjacent outer angles.
The sum of any two sides of a triangle is greater than the third side, and the difference between any two sides is equal to the third side (converted into inequality, we can get such a simple inequality group, A+B >; c，a+c & gt； b，b+ c & gt； A, a, b and c are the lengths of three sides of a triangle)
3*, congruent triangles
The nature of congruent triangles: the corresponding angles of congruent triangles are equal, and the corresponding sides are equal.
Congruent triangles's judgment: ① S.A.S2A.S.A3A.A.S4S.S5H.L
Congruent triangles's knowledge points are not the focus of IMC multiple-choice questions. In IMC competition, 25 questions are multiple-choice questions. We only need to focus on understanding the nature of congruent triangles and pay attention to the equal sides. When two triangles are found to be identical, they can be used directly without proof, saving time for the following problems.
IMC's multiple-choice questions did not consider congruent triangles's knowledge points. In IMC competition, 25 questions are multiple-choice questions. We only need to focus on understanding the nature of congruent triangles and pay attention to the equal sides. Follow-up questions save time.
4, isosceles triangle, equilateral triangle, right triangle
An isosceles triangle has two equal waists and two equal base angles.
The three lines of the isosceles triangle are one: the height on the bottom of the isosceles triangle, the median line on the bottom and the bisector of the vertex angle coincide with each other.
Three sides of an equilateral triangle are equal, and three internal angles are equal to 60.
Determination of equilateral triangle: An isosceles triangle with an internal angle equal to 60 is an equilateral triangle.
The two acute angles of a right triangle are complementary.
The sum of squares of two right-angled sides of a right-angled triangle is equal to the square of the hypotenuse (Pythagorean 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 (the inverse theorem of Pythagorean theorem).
An isosceles triangle has two equal sides and equal base angles.
The three lines of an isosceles triangle are integrated: the height, the median line and the bisector of the vertex of the isosceles triangle coincide with each other.
Three sides of an equilateral triangle are equal, and three internal angles are equal to 60.
Judgment of equilateral triangle: An isosceles triangle with an internal angle equal to 60 is an equilateral triangle.
The two acute angles of a right triangle are complementary.
The sum of squares of two right-angled sides of a right-angled triangle is equal to the square of the hypotenuse (Pythagorean theorem).
If the sum of squares of two sides of a triangle is equal to the square of the third side, the triangle is a right triangle (the inverse of Pythagorean theorem).
In the exam, when you see a regular hexagon and a square, you must associate it with a regular triangle and an isosceles right triangle respectively (a regular hexagon can be simply decomposed into six small regular triangles, and a square can be regarded as two isosceles right triangles). Moreover, students need to keep in mind the proportional relationship between the sides of 30, 60 and 90 right-angled triangles (special values in sine and cosine) and the proportional relationship between the sides of isosceles right-angled triangles (if you have forgotten it, please use Pythagorean theorem to deduce it yourself to deepen your impression).
In a word, the geometry problem is a very important part in the IMC competition exam. I hope that students can combine the exam questions in Pastepaper and review more in order to get high marks in the exam on February 2 -3!