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The method of drawing auxiliary line in the proof of congruent triangle
Try to find the most representative lines, such as vertical bisector, angular bisector and so on. Different graphics draw auxiliary lines in different ways. You have to do more practice to improve
Tips for making auxiliary lines of mathematical congruent triangles
If there is a midpoint, first consider the median line, and then construct an congruent triangle. The ideal situation is to construct a common angle or a common edge. If the triangle has a right angle, it is preferred to make a vertical line... Too much to say. It is suggested to buy a Xue Jinxing junior high school mathematics basic knowledge manual, very complete
Congruent triangle in grade two In △ ABC, ∠ ACB is a right angle, ∠ B = 60 °, ad and CE are bisectors of ∠ BAC and ∠ BCA respectively, and AD and CE intersect at point F. it is proved that Fe = FD
It is proved that FM ⊥ BC in M, FN ⊥ AB in n
∵∠B=60°
∴∠MFN=120°
∵ ad, CE are angular bisectors
∴FM=FN
∠FAC+∠FCA=15°+45°=60°
∴∠AFC=120°
∴∠EFD=120°
∴∠EFN=∠DFM
∵FE=FM,∠FNE=∠FMD
∴△FEN≌△FMD
∴FD=FE
Exercises on congruent triangles Help out a few exercises, 8 fill in the blanks, 5 proof questions are all about congruence! If the difficulty is medium, you can add it to see how it is
Answer: 5154225 - Magic apprentice level 1 7-29 15:37
There is and only one straight line after two
2. The line segment between two points is the shortest
3 the complementary angles of the same angle or equal angle are equal
The remainder of the same angle or equal angle is equal
There is and only one line perpendicular to the known line
Among all the lines connected by a point outside the line and the points on the line, the vertical line is the shortest
The parallel axiom passes through a point outside the line, and there is only one line parallel to this line
If both lines are parallel to the third line, they are parallel to each other
The two lines are parallel
The internal staggered angle is equal and the two straight lines are parallel
The inner angle of the same side is complementary, and the two straight lines are parallel
The two lines are parallel and the same angle
The two straight lines are parallel and the angle of internal stagger is equal
The two straight lines are parallel and complementary to each other
Theorem 15 the sum of the two sides of a triangle is greater than the third side
16 infer that the difference between the two sides of the triangle is less than the third side
The sum of the three interior angles of a triangle is equal to 180 degrees
Inference 1 two acute angles of a right triangle are complementary
Inference 2 an outer angle of a triangle is equal to the sum of two interior angles that are not adjacent to it
Inference 3 an outer angle of a triangle is greater than any inner angle that is not adjacent to it
The corresponding sides and angles of an congruent triangle are equal
There are two congruent triangles whose angles are equal to each other
The 23 angle angle axiom (ASA) has two congruent triangles corresponding to two angles and their clamped edges
24 corollary (AAS) there are two congruent triangles corresponding to the opposite sides of one of the two angles
The 25 side side side axiom (SSS) has three sides corresponding to the congruence of two triangles
The axiom of hypotenuse and right angled side (HL) is the congruence of two right triangles with a hypotenuse and a right angle
Theorem 1 the distance from a point on the bisector of an angle to both sides of the angle is equal
Theorem 2 a point with the same distance from both sides of an angle is on the bisector of the angle
The bisector of an angle 29 is the collection of all points equidistant from both sides of the angle
Property theorem of isosceles triangle two base angles of isosceles triangle are equal
Corollary 1 the bisector of the vertex angle of an isosceles triangle bisects and is perpendicular to the base
The bisector of the top angle, the center line on the bottom edge and the height on the bottom edge of an isosceles triangle coincide with each other
Corollary 3 the angles of an equilateral triangle are equal, and each angle is equal to 60 degrees
Theorem of isosceles triangle if two angles of a triangle are equal, the opposite sides of the two angles are equal (equal angle to equal side)
Corollary 1 a triangle whose three angles are equal is an equilateral triangle
Corollary 2 an isosceles triangle with an angle equal to 60 ° is an equilateral triangle
In a right triangle, if an acute angle is equal to 30 degrees, the right angle it faces is equal to half of the hypotenuse
The center line on the hypotenuse of a right triangle is equal to half of the hypotenuse
Theorem 39 the distance between a point on the vertical bisector of a line segment and its two endpoints is equal
An inverse theorem and a point on the vertical bisector of a line segment whose two endpoints are equidistant
The vertical bisector of line 41 can be regarded as the set of all points equal to the distance between the two ends of the line
Theorem 1 two symmetrical figures about a straight line are holomorphic
Theorem 2 if two figures are symmetric about a straight line, then the axis of symmetry is the vertical bisector of the line connecting the corresponding points
Theorem 3 two figures are symmetric about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the symmetry axis
45 inverse theorem if the line of corresponding points of two figures is vertically bisected by the same line, then the two figures are symmetrical about the line
Pythagorean theorem the sum of squares of two right sides a and B of a right triangle is equal to the square of hypotenuse C, that is, a ^ 2 + B ^ 2 = C ^ 2
47 inverse theorem of Pythagorean theorem if the three sides of a triangle a, B, C are related, a ^ 2 + B ^ 2 = C ^ 2, then the triangle is a right triangle
Theorem 48 the sum of interior angles of quadrilateral is equal to 360 degrees
The sum of the exterior angles of 49 quadrilateral is equal to 360 degrees
Theorem of sum of interior angles of 50 polygons the sum of interior angles of n-polygon is equal to (n-2) × 180 degrees
It is inferred that the sum of the exterior angles of any polygon is equal to 360 degrees
Property theorem of parallelogram 1 diagonal equality of parallelogram
53 property theorem of parallelogram 2. Opposite sides of parallelogram are equal
54 infer that the parallel line segments sandwiched between two parallel lines are equal
55 property theorem of parallelogram 3 the diagonals of parallelogram bisect each other
Theorem 1 two groups of quadrilateral with equal diagonal angles are parallelograms
Theorem 2 two groups of quadrilateral whose opposite sides are equal are parallelogram
58 parallelogram judgment theorem 3 a quadrilateral whose diagonals are bisected is a parallelogram
Theorem 4 a group of parallelograms whose opposite sides are equal are parallelograms
Theorem 1 the four corners of a rectangle are right angles
Theorem 2 the diagonals of rectangles are equal
Theorem 1 a quadrilateral with three right angles is a rectangle
Theorem 2 a parallelogram with equal diagonals is a rectangle
Theorem 1 four sides of a diamond are equal
Theorem 2 the diagonals of diamond are perpendicular to each other, and each diagonal is divided into a group of diagonals
66 diamond area = half of diagonal product, i.e. s = (a × b) △ 2
Theorem 1 a quadrilateral whose four sides are equal is a diamond
68 diamond judgment theorem 2 the parallelogram whose diagonals are perpendicular to each other is a diamond
Theorem 1 the four corners of a square are right angles and all four sides are equal
Theorem 2 two diagonals of a square are equal and bisect each other vertically
Theorem 1 two graphs of central symmetry are congruent
Theorem 2 for two graphs of centrosymmetry, the line of symmetric points passes through the center of symmetry and is bisected by the center of symmetry
73 inverse theorem if the line of the corresponding points of two graphs passes through a certain point and is bisected by this point, then the two graphs are symmetrical about this point
Property theorem of isosceles trapezoid two angles of isosceles trapezoid on the same base are equal
The two diagonals of an isosceles trapezoid are equal
76 isosceles trapezoid judgment theorem a trapezoid with two equal angles on the same base is an isosceles trapezoid
A trapezoid with equal diagonals is an isosceles trapezoid
A line segment that is cut off by a group of parallel lines on a straight line
If it is equal, then the segments cut on other lines are equal
Inference 1 a straight line passing through the middle point of one trapezoid waist and parallel to the bottom of one waist will bisect the other waist
Inference 2 a straight line passing through the midpoint of one side of a triangle and parallel to the other side must bisect the third side
Theorem of the median line of a triangle the median line of a triangle is parallel to the third side and equal to half of it
82 trapezoid median line theorem the median line of a trapezoid is parallel to two bottoms and is equal to half of the sum of two bottoms L = (a + b) △ 2 s = l × H
The basic properties of 83 (1) ratio if a: B = C: D, then ad = BC
If ad = BC, then a: B = C: D WC 呁 / s ﹥ x1e?
84 (2) if a / b = C / D, then (a ± b) / b = (C ± d) / d
85 (3) isometric property if a / b = C / D = =m／n(b+d+… +N ≠ 0)
(a+c+… +m)／(b+d+… +n)=a／b
The theorem of proportionality of parallel line segments: three parallel lines cut two straight lines, and the corresponding line segments obtained are proportional
A straight line parallel to one side of the triangle cuts off the other two sides (or the extension of both sides), and the corresponding line segment is proportional
Theorem 88 if a line cuts the two sides of a triangle (or the extension of both sides) in proportion, then the line is parallel to the third side of the triangle
A straight line parallel to one side of the triangle and intersecting with the other two sides. The three sides of the triangle cut are proportional to the three sides of the original triangle
Theorem 90 If a straight line parallel to one side of a triangle intersects the other two sides (or the extension lines of both sides), the triangle formed is similar to the original triangle
Theorem 1 two angles are equal and two triangles are similar (ASA)
Two right triangles divided by the height of the hypotenuse are similar to the original triangle
93 judgment theorem 2 two sides are proportional and the included angles are equal, two triangles are similar (SAS)
94 judgment theorem 3 three sides are proportional and two triangles are similar (SSS)
As shown in the picture, there are two flagpoles on the square, which are placed perpendicular to the ground. It is known that the sunlight AB and de are parallel. After measuring the shadow of the two flagpoles in the sunlight, are the two flagpoles of the same height? Tell me your reasons
The two flagpoles are equal in height
The reasons are as follows: ∵ the sun's rays AB and de are parallel,
∴∠B=∠E，
∵ both flagpoles are placed perpendicular to the ground,
∴∠C=∠F=90°，
∵ the shadow of the two flagpoles is the same length in the sunlight,
∴BC=EF，
In △ ABC and △ def,
∠B＝∠E
BC＝EF
∠C＝∠F ，
∴△ABC≌△DEF（ASA），
∴AC=DF，
That is, the height of the two flagpoles is equal
As shown in the figure, in RT △ ABC, ab = AC, ∠ BAC = 90 °, 1 = ∠ 2, CE ⊥ BD is extended to e. it is proved that BD = 2ce
It is proved that: extended CE and Ba intersect at point F, as shown in the figure, ∵ be ⊥ EC, bef = ∠ CEB = 90 °. ? BD bisection ? ABC, ? 1 = ∠ 2, ∵ f = ∠ BCF, ? BF = BC, ∵ be ⊥ CF, ? CE = 12CF, ? ABC, AC = AB, ∵ a = 90 °, CBA = 45 °, f = (180-45) ⊥ 2 = 67.5 °
Exercises on congruent triangles! Point C is on BD, AC is perpendicular to BD, BD is perpendicular to point C, be is ad perpendicular to point E, CF = CD, then are ad and BF equal? Why
Because AC is perpendicular to BD and be is perpendicular to ad, triangle ACD and triangle BCF are right triangles. And because CF = CD, triangle ACD and triangle BCF are congruent (two corners and sides are equal respectively). Therefore, ad and BF are equal
One angle of an isosceles triangle is 110 ° and its other two angles are______ One angle of an isosceles triangle is 80 ° and its other two angles are______ .
① When 110 ° is the top angle, the base angle = (180 ° - 110 °) / 2 = 35 °, when 110 ° is the base angle, another base angle is also 110 ° because 110 ° + 110 ° is more than 180 ° so it does not conform to the triangle interior angle sum theorem, that is, it can not form a triangle
One angle of an isosceles triangle is 110 ° and its other two angles are______ One angle of an isosceles triangle is 80 ° and its other two angles are______ .
① When 110 ° is the top angle, the base angle = (180 ° - 110 °) / 2 = 35 °,
When 110 ° is the base angle, the other base angle is also 110 °,
Because 110 degree + 110 degree > 180 degree, it does not conform to the triangle angle sum theorem
② This question can be divided into two situations
When the 80 ° angle is the top angle, the base angle = (180 ° - 80 °) / 2 = 50 °; then the other two angles are 50 ° and 50 ° respectively
When the 80 ° angle is the base angle, the top angle is 180 ° - 2 × 80 ° = 20 ° and the other two angles are 20 ° and 80 ° respectively
Therefore, the answer is: 35 ° 35 ° and 50 ° 50 ° or 20 ° 80 ° respectively
It is proved that a triangle with two equal angles is an isosceles triangle, and an congruent triangle is constructed by making the base line of the triangle As the title It seems that it is not possible to side, edge and edge. The bottom edge is equal, and the middle line is a common side. But the title does not say that this is an isosceles triangle (isosceles triangle is required to prove). How do you know that the remaining edge is equal?
By the way, make the center line of the bottom edge. After making the bottom edge, two triangles on both sides have two sides equal and one angle is equal. The reason why an edge angle can't be proved is because one triangle is an acute angle triangle and the other is an obtuse angle triangle, One is an obtuse angle) and the corresponding sides can be guaranteed to be the same. (you can put the acute angle triangle inside the obtuse angle triangle). The two corners of the bottom edge are exactly the same, so their corresponding sides, that is, the waist, must be equal
RELATED INFORMATIONS
- 1. Are two right triangles with an angle of 60 ° congruent? Are two isosceles triangles congruent with an obtuse angle
- 2. In isosceles △ ABC, ab = AC = 13cm, BC = 10cm. Find the radius of the circumcircle of isosceles △ ABC
- 3. If the circumference of the isosceles △ ABC is 10 cm, the bottom edge length is y cm, and the waist length is x cm, then the value range of waist length x is______ .
- 4. In the triangle ABC, ab = AC, the center line be of triangle ABC divides the circumference of triangle into two parts of 9cm and 12cm, and calculates the length of BC in triangle ABC You have to draw your own pictures, The length of the center line is also included in the perimeter of the two parts
- 5. Given that the circumference of the isosceles triangle ABC is 40cm, ad is the height on the bottom edge, and the circumference of the triangle abd is 30cm, what is the length of ad
- 6. As shown in the figure, in the isosceles triangle ABC, ab = AC, point D is a point on BC, de ∥ AC intersects AB at point E, DF ∥ AB intersects AC at point F, ① As shown in Figure 1, when point D is on BC, there is de + DF = AB, please explain the reason. ② as shown in Figure 2, when point D is on the extension line of BC, please refer to figure 1 to draw the correct figure, write down the relationship between De, DF and AB, and write the proof process,
- 7. It is known that Mn is the vertical bisector of line AB, and C and D are two points on Mn (1) Δ ABC, △ abd are isosceles triangles; （2）∠CAD=∠CBD．
- 8. As shown in the figure, given that AB is 2cm longer than AC, the vertical bisector of BC intersects AB at D, crosses BC at e, and the circumference of △ ACD is 14cm, then ab=______ cm，AC=______ cm．
- 9. As shown in the figure: the circumference of △ ABC is 24cm, ab = 10cm, the vertical bisector De of edge AB intersects BC, the edge is at point E, and the perpendicular foot is D, calculate the circumference of △ AEC
- 10. As shown in the figure, it is known that in trapezoidal ABCD, BD bisection ∠ ABC, ∠ a = 120 ° and BD = BC = 4 3, then s trapezoid ABCD = () A. 4 Three B. 12 C. 4 3-12 D. 4 3+12
- 11. Can I divide two triangles with 24 ', 84' and 104 ', 52' angles into two isosceles triangles with a straight line (if yes, please write down the divided? B If you can, write the top angle readings of the two isosceles triangles
- 12. The area and perimeter of an congruent triangle are equal. Are the two triangles with the same area and perimeter congruent triangles?
- 13. The isosceles triangle ABC with base BC is still isosceles triangle after being divided into two smaller triangles by a line passing through a vertex Draw all the situations and mark the angles
- 14. In the triangle ABC, ab = AC, the line passing through a vertex intersects the opposite side of D. the line divides the original triangle into two isosceles triangles In the triangle ABC, ab = AC, the line passing through a vertex intersects the opposite side of D. the line divides the original triangle into two isosceles triangles. Find the degree of the three inner angles of the triangle ABC.
- 15. The degree of the top angle of an isosceles triangle is four times that of the base angle. How many degrees are the base angle and the top angle of the isosceles triangle? What triangle is this?
- 16. If an interior angle of an isosceles triangle is 80 degrees, how many degrees or degrees does the top angle differ from the base angle?
- 17. For an isosceles triangle with a vertex angle of 30 degrees, if the waist length is 2, then the height () on the waist is () and the area of the triangle is ()
- 18. The top angle of an isosceles triangle is 43 degrees. What is its base angle?
- 19. 1. Please analyze Mr. Wang's behavior according to the material questions. 2. If you are Mr. Wang, This material is entitled pedagogy case: Mr. Wang was in class when he heard the music coming from the game machine behind the third group. Mr. Wang quickly walked to the back and looked down to find that it was Lu's game machine. Mr. Wang held out his hand and put Lu's game machine away. It looked like it was an advanced game machine with a card inserted. After returning to the platform, Mr. Wang said, "if you take it, you won't return it, If you ask your parents to take it. "Lu was not happy to hear this, he replied," why do you accept me? Other people have fun. I just accidentally pressed it. "Mr. Wang was even more angry when he heard this. He said," the teacher will accept you. You are not lucky. If you answer back, the teacher will smash the machine. "Lu was angry and rushed to the podium and snatched the machine back, So Mr. Wang tried to snatch the machine back from his hand. Lu was so angry that he made a big noise on his seat. Mr. Wang was so angry that he had to pull Lu to the front to punish him 1. Please analyze Mr. Wang's behavior according to the material questions 2. If you are Mr. Wang,
- 20. How many pieces of triangular towel can be made with 6.4 m long and 1.6 m wide cloth when making two pieces of bandage with right angle sides of 0.8 m?