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.
Quick judgment method: 1) equilateral triangle: the sum of the smallest two sides is greater than the third side to form a triangle. 2) isosceles triangle: if the sum of the two waists is greater than the bottom, a triangle can be formed. 3) equilateral triangle: it can definitely be composed.
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. Drawing method of high line, middle line and angular 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 the triangle and the extension line of the other side is called the external angle of the triangle (the principle of three out of six).
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.
First, the basic multiple choice questions
1. The lengths of two sides of the triangle are 2 and 6, and the third side is even. Then the circumference of this triangle is ().
10 b . 12 c . 14d . 16
2.In △ABC，AB = 4a，BC = 14，AC = 3a。 Then the range of a is ().
a . a > 2 b . 2 < a < 14 c . 7 < a < 14d . a < 14
3. Among the three internal angles of a triangle, the number of acute angles is at least ().
A.0 B. 1 C.2 D.3
4. The following statement is wrong ()
A. the three bisectors of a triangle intersect at a point B. The three median lines of a triangle intersect at one point.
C. The three heights of a triangle intersect at a point D. The straight lines of the three heights of a triangle intersect at a point.
5. The line segment that can divide a triangle into two triangles with equal area is ().
A. center line B. angle bisector C. height line D. angle bisector of triangle
6. As shown in Figure 5- 12, it is known that ∠ ACB = 90, CD⊥AB, and the vertical foot is D, then the angle equal to ∠ A in the figure is ().
A.∠ 1 B. ∠2 C. ∠ B. ∠ 1, ∠2 and ∠ B.
7. If point P is any point in △ABC, then the relationship between ∠APC and ∠B is () A. ∠ APC > ∠ B. ∠ APC = ∠ B.C. ∠ APC < ∠ B. It cannot be determined.
8. Given that a, b and c are the length of △ABC, and m = (A+B+C) (A+B-C) (A-B-C), then ().
A.m > 0 b.m = 0 c.m < 0 d. Not sure.
Second, fill in the blanks
1. The lengths of the five line segments are 1, 2, 3, 4, 5 respectively, and any three of them can be _ _ _ _ _ triangles.
2. In △ABC, AB = 6, AC = 10, Then the range of BC edge is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
3. The ratio of the degrees of the three internal angles of a triangle is 2: 2: 1, and this triangle is _ _ _ _ _ _ _.
4. If the lengths of two sides of an isosceles triangle are 15cm and 7cm respectively, the perimeter of the triangle is _ _ _ _ _ _ _ _.
6. In a right triangle, if the difference between two acute angles is 40, then the degrees of these two acute angles are _ _ _ _ _ _.
7. In △ABC, ∠ A-∠ B = 30, ∠ C = 4 ∠ B, then ∠ C = _ _ _ _ _ _
8. As shown in Figure 5- 13, in △ABC, AD⊥BC, GC⊥BC, CF⊥AB, BE⊥AC, the vertical feet are D, C, F and E respectively, so _ _ _ _ is BC in △ABC.
9. As shown in Figure 5- 14, the bisectors of the two external angles of △ABC intersect at point D. If ∠ A = 50, ∠ D = _ _ _ _.
10. As shown in Figure 5- 15, in △ABC, ∠ A = 60, and the bisectors BD and CD of ∠ABC and ∠ACB intersect at point D, then ∠ BDC = _ _ _ _
12. If the circumference of an isosceles triangle is 24cm and the waist length is xcm, then the value range of x is _ _ _ _ _ _.
Third, expand multiple-choice questions
1. The line segment that must be within △ABC is ()
A three heights, three bisectors and three median lines of an acute triangle
B three heights, three median lines and an angular bisector of an obtuse triangle
One midline, two bisectors and three heights of an arbitrary triangle.
D. Three heights, three bisectors and three median lines of a right triangle.
2. In the following statement, the correct one is ()
A. An obtuse triangle must not be an isosceles triangle or an equilateral triangle.
B. An isosceles triangle must be an acute triangle or a right triangle.
C. A right triangle must not be an isosceles triangle or an equilateral triangle.
An equilateral triangle must not be an obtuse triangle or a right triangle.
3. As shown in the figure, in △ABC, d and e are two points on BC respectively, and BD = de = EC, then there are () A.4 pairs of B.5 pairs of C.6 pairs of D.7 pairs of triangles with equal areas.
4. If the intersection of three heights of a triangle happens to be a vertex of the triangle, then the triangle is ().
A. acute triangle B. obtuse triangle C. right triangle D. uncertainty
5. The three line segments given in the following questions cannot form a triangle ().
A.A+ 1, A+2, a+3 (a > 0) B. The ratio of the three line segments is 4∶6∶ 10.
C.3cm，8cm， 10cm D.3a，5a，2a+ 1(a>0)
6. If one side of an isosceles triangle is 7 and the other side is 4, then the circumference of the isosceles triangle is ().
A.18b.15c.18 or 15d. Not sure.
7. The two sticks are 5cm and 7cm respectively. Choose the third stick and nail them into a triangle. If the length of the third stick is even, the third stick has () values.
a3 b . 4 c . 5d . 6
10. The sum of all external angles of the triangle is () A.180b.360c.720d.540.
1 1. In an acute triangle, the range of angle α is ().
a . 0