Calculating with triangles


Calculating with triangles

Tính toán liên quan đến tam giác


The base angles of an isosceles triangle are equal.


If a = b, then x = y

If the base angles of a triangle are equal, the triangle is isosceles.


If x = y, then a = b

The measure of an exterior angle is equal to the sum of the measures of the remote interior angles.


l is a straight line.

Then, x = y + z

In a triangle, the greater angle lies opposite the greater side.


If a < b, then y < x


If a < b, then y < x

Similar Triangles


If ΔABC ~ ΔDEF, then

m∠A = m∠B

m∠B = m∠E

m∠C = m∠F

 and \frac{a}{d}=\frac{b}{e}=\frac{c}{f} 


The sum of the interior angles of a triangle is 180 degrees.

m∠A + m∠B + m∠A  = 180º

The area of a triangle is one-half the product of the altitude to a side and the side.

Area of △ABC = \frac{ADxBC}{2}


Note: If m∠A = 90º

Area also = \frac{ADxBC}{2}

In a right triangle


c2 = a2 + b2

and x°+ y° = 90°

Memorize the following standard triangles


The sum of the lengths of two sides of a triangle is greater than the length of the third side. (This is like saying that the shortest distance between two points is a straight line.)


a + b > c

a + c > b

b + c > a

Example 1

In the diagram below, what is the value of x?


(A) 20

(B) 25

(C) 26

(D) 45

(E) 48


Choice C is correct.

Method 1: Use right triangle. Then,

x2 = 242 + 102

    = 576 + 100

    = 676

Thus, x = 26 (Answer)

Method 2: Notice that ΔMNP is similar to one of the standard triangles:


This is true because

12/24 = 5/10

Hence, 12/24 =13/x or x = 26 (Answer)

Example 2

If Masonville is 50 kilometers due north of Adamston and Elvira is 120 kilometers due east of Adamston, then the minimum distance between Masonville and Elvira is

(A) 125 kilometers

(B) 130 kilometers

(C) 145 kilometers

(D) 160 kilometers

(E) 170 kilometers


Choice B is correct. Draw a diagram first.


The given information translates into the diagram above.

The triangle above is a multiple of the special 5–12–13 right triangle.

50 = 10×5

120 = 10×12

Thus, x = 10×13 = 130 kilometers

(Note: The Pythagorean Theorem could also have been used: 502 + 1202 = x2.)

Example 3

In triangle ABC, if a > c, which of the following is true?


(A) BC = AC

(B) AB > BC

(C) AC > AB

(D) BC > AB

(E) BC >AC


Choice D is correct.

From basic geometry, we know that, since m∠BAC > m∠BCA, then leg opposite BAC > leg opposite BCA, or BC > AB

Example 4

The triangle above has side BC = 10, angle B = 45°, and angle A = 90°. The area of the triangle


(A) is 15

(B) is 20

(C) is 25

(D) is 30

(E) Cannot be determined.


Choice C is correct.

First find angle C.

90° + 45° + m∠C = 180°

So m∠C = 45°.

We find AB = AC, since m∠B = m∠C = 45°.

Since our right triangle ABC has BC = 10, (the right triangle \frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},1) multiply by 10 to get a right triangle: \frac{10\sqrt{2}}{2}, \frac{10\sqrt{2}}{2},10

Thus side AB = \frac{10\sqrt{2}}{2}={5\sqrt{2}}  

         side AC = \frac{10\sqrt{2}}{2}={5\sqrt{2}}

Now the area of triangle ABC is \frac{5\sqrt{2}x5\sqrt{2}}{2}=\frac{25x2}{2}=25

Example 5

In the figure above, what is the value of x?


(A) 30

(B) 40

(C) 50

(D) 80

(E) 100


Choice B is correct.

Remember triangle facts. Use Statement II.

ADB is an exterior angle of ΔACD, so mADB = x + x = 2x          (1)

In ΔADB, the sum of its angles = 180, so     

mADB + 55 + 45 = 180

or   mADB + 100 = 180

or             mADB = 80                                                                   (2)

Equating (1) and (2) we have 2x = 80, x = 40 (Answer)

Example 6

Which of the following represents all of the possibilities for the value of a in the figure above?


(A) 1 < a < 9

(B) 4 < a < 5

(C) 0 < a < 9

(D) 4 < a < 9

(E) 5 < a < 9


Choice A is correct. Since the sum of the lengths of two sides of a triangle is greater than the length of the third side, we have:

a + 5 > 4      (1)

a + 4 > 5      (2)

5 + 4 > a      (3)

From (2) we get: a > 1.

From (3) we get: 9 > a.

This means that 9 > a > 1, or 1 < a < 9.


Properties of triangles – Part 2


Các thuộc tính của tam giác – Phần 2

similar: Đồng dạng

congruent: Tương đẳng

median: Đường trung tuyến

vertex: Đỉnh (số ít), vertices (số nhiều)

midpoint: Điểm giữa.

opposite side: Cạnh đối diện

cross: Cắt nhau, giao nhau, đi qua

angle bisectors: Các góc phân giác

center: Tâm

inscribed: Nội tiếp

circumscribed: Ngoại tiếp

altitude: Độ cao, đường cao

perpendicular bisectors: Đường trung trực

inequality: Bất đẳng thức


Similar triangles

Two triangles are said to be similar (having the same shape) if their corresponding angles are equal. The sides of similar triangles are in the same proportion. The two triangles below are similar because they have the same corresponding angles.


a : d = b : e = c : f


Two triangles both have angles of 30°, 70°, and 80°. If the sides of the triangles are as indicated below, find the length of side x.


The two triangles are similar because they have the same corresponding angles.
The corresponding sides of similar triangles are in proportion, so x : 3 = 6 : 4. This can be rewritten as \frac{x}{3}=\frac{6}{4}. Multiplying both sides by 3 gives x=\frac{18}{4} , or x=4\tfrac{1}{2}.

Congruent triangles

Two triangles are congruent (identical in shape and size) if any one of the following conditions is met:

1. Each side of the first triangle equals the corresponding side of the second triangle.

2. Two sides of the first triangle equal the corresponding sides of the second triangle, and their included angles are equal. The included angle is formed by the two sides of the triangle.

3. Two angles of the first triangle equal the corresponding angles of the second triangle, and any pair of corresponding sides are equal.


Triangles ABC and DEF in the diagrams below are congruent if any one of the following conditions can be met:


1. The three sides are equal

(sss) = (sss)


2. Two sides and the included angle are equal

(sas) = (sas)


3. Two angles and any one side are equal

(aas) = (aas) or (asa) = (asa)



In the equilateral triangle below, line AD is perpendicular (forms a right angle) to side BC. If the length of BD is 5 feet, what is the length of DC?


Since the large triangle is an equilateral triangle, each angle is 60°.

Therefore ∠B is 60° and ∠C is 60°.

Thus, ∠B = ∠C. ADB and ADC are both right angles and are equal.

Two angles of each triangle are equal to the corresponding two angles of the other triangle.

Side AD is shared by both triangles and side AB = side AC.

Thus, according to condition 3 “Two angles and any one side are equal”, the two triangles are congruent.

Then BD = DC and, since BD is 5 feet, DC is 5 feet.


The medians of a triangle are the lines drawn from each vertex to the midpoint of its opposite side. The medians of a triangle cross at a point that divides each median into two parts:

  • One part of one-third the length of the median.
  • The other part of two-thirds the length.

Angle bisectors

The angle bisectors of a triangle are the lines that divide each angle of the triangle into two equal parts. These lines meet in a point that is the center of a circle inscribed in the triangle.



The altitudes of the triangle are lines drawn from the vertices perpendicular to the opposite sides. The lengths of these lines are useful in calculating the area of the triangle, since the area of the triangle is 1/2(base)(height), and the height is identical to the altitude.


Perpendicular bisectors

The perpendicular bisectors of the triangle are the lines that bisect and are perpendicular to each of the three sides. The point where these lines meet is the center of the circumscribed circle.


The sum of any two sides of a triangle is greater than the third side.

If the three sides of a triangle are 4, 2, and x, then what is known about the
value of x?


Since the sum of two sides of a triangle is always greater than the third side, then:

4 + 2 > x

4 + x > 2

2 + x> 4

These three inequalities can be rewritten as

6 > x

x > -2

x > 2

For x to be greater than -2 and 2, it must be greater than 2.
Thus, the values of x are 2 < x < 6.

Tải bài viết

Angles Practice 1

  • To convert degrees to radians, multiply the degree measure by  \frac{\pi radians}{180^{0}}.


Convert 85° to radians:


85^{^{0}}=85(\frac{\\\pi radians}{180^{0}})=\frac{17\pi }{36}

  • To convert from radians to degrees, multiply the radian measure by \frac{180^{0}}{\pi radians}.


Convert \frac{\pi }{6} radians to degrees.


\frac{\pi }{6}=\left ( \frac{\pi radians}{6} \right )\left (\frac{180^{0}}{\pi radians} \right )=30^{^{0}}

1. In the figure below, angles 3m and 4n are vertical angles. What is the value of m in terms of n?


(A) n

(B) 12n

(C) \frac{n}{12}

(D) \frac{3n}{4}

(E) \frac{4n}{3}


The two angles are vertical angles, so their measures are equal:

3m = 4n

To solve for m, divide both sides by 3:


Thus, the right answer is (E)

2. In the figure below, v = 3u. What is the value of w?


(A) 15

(B) 30

(C) 45

(D) 60

(E) 75


Angles u and v are supplementary angles because they complete a line, so u + v = 180.

Substitute 3u for v into this equation and solve for u:

u + 3u = 180

4u = 180

u = 45

The two angles u and w are vertical angles, so they’re equal. Thus, w = 45, and the right answer is (C) .

3. In the following diagram, angle 1 is equal to 40°, and angle 2 is equal to 150°. What is the number of degrees in angle 3?


(A) 70°

(B) 90°

(C) 110°

(D) 190°

(E) The answer cannot be determined from the given information.


Choice C is correct. In the problem it is given that ∠1 = 40° and ∠2 = 150°. The diagram below makes it apparent that:

(1) ∠1 = ∠4 and ∠3 = ∠5 (vertical angles);

(2) ∠6 + ∠2 = 180° (straight angle);

(3) ∠4 + ∠5 + ∠6 = 180° (sum of angles in a triangle).

To solve the problem, ∠3 must be related through the above information to the known quantities in ∠1 and ∠2.

Proceed as follows:

∠3 = ∠5, but ∠5 = 180° – ∠4 – ∠6.

∠4 = ∠1 = 40° and ∠6 = 180° – ∠2 = 180° – 150° = 30°.

Therefore, ∠3 = 180° – 40° – 30° = 110°.


4. In the figure below, angles p and q are complementary, with 2p = 3q. What is the measure of angle p?


The problem states that angles p and q are complementary, so you have two equations to work with:

p + q = 90

2p = 3q

Solve the top equation for q:

q = 90 – p

Now substitute 90 – p for q in the second equation and solve for p:

2p = 3(90 – p)

2p = 270 – 3p

5p = 270

p = 54

Therefore, the answer is

————End of Angles Practice 1————

Vocabulary Angles


Các từ vựng về độ đo các góc

plane: Phẳng

geometry: Hình học

plane geometry: Hình học phẳng

point: Điểm

line: Đường thẳng

ray: Tia, vector

length: Chiều dài

straight: Thẳng; straight angle: Góc bẹt

curved: Cong

segment: Đoạn.

distance: Khoảng cách

angle: Góc

degree: Độ

side: Cạnh

parallel: Song song

crossed: Cắt, cắt bởi

transversal: Đường ngang, đường hoành

diagram: Hình vẽ, hình minh họa

circle: Đường tròn, hình tròn

half: Một nửa

a half circle: Nửa đường tròn

greater than: Lớn hơn

less than: Nhỏ hơn

equal, equal to: Bằng, bằng nhau

intersect: Giao

union: Hợp, hội

measure: Số đo

classified: Phân loại

Một số từ kết hợp với từ khác hoặc có tên gọi trong lĩnh vực toán học

acute: Nhọn

right: Vuông.

obtuse: Tù

reflex: Phản.

complementary: Bù, bù nhau.

supplementary: Phụ, phụ nhau.

vertical: Đỉnh.


  • Angles. An angle is formed when two lines intersect at a point.

Angle B,

Angle ABC,

∠B, and ∠ABC are all possible names for the angle shown.

  • The measure of the angle is given in degrees.
  • If the sides of the angle form a straight line, then the angle is said to be a straight angle and has 180°.
  • A circle has 360°, and a straight angle is a turning through a half circle.
  • All other angles are either greater or less than 180°.

Angles are classified in different ways:

  • An acute angle has less than 90°.
  • A right angle has exactly 90°.
  • An obtuse angle has between 90° and 180°.
  • A straight angle has exactly 180°.
  • A reflex angle has between 180° and 360°.
  • Complementary angles. Two angles are complementary if their sum is 90°.

For example, an angle of 49° and an angle of 41° are complementary.

  • Supplementary angles . Two angles are supplementaryif their sum is 180°. If one angle is 120°, then its supplement is 60°.
  • Vertical angles (các góc đối đỉnh) . These are pairs of opposite angles formed by the intersection of two straight lines. Vertical angles are always equal to each other.

Example: In the diagram shown, angles AEC and BED are equal because they are vertical angles. For the same reason, angles AED and BEC are equal.

When two parallel lines are crossed by a third straight line (called a transversal), then all the acute angles formed are equal, and all of the obtuse angles are equal.



In the diagram below, angles 1, 4, 5, and 8 are all equal. Angles 2, 3, 6, and 7 are also equal.

——End of Vocabulary Angles———-

Properties of triagles – Part 1


property: Thuộc tính

properties of triangles: Các thuộc tính của tam giác

line segment: Đoạn thẳng.

closed figure: Hình khép kín.

The sum of the angles of a triangle: Tổng các góc của một tam giác.

scalene: Lệch, không cân, không đều cạnh.

scalene triangles: Các tam giác thường.

unequal side: Cạnh không bằng nhau.

hypotenuse: Cạnh huyền.

Pythagorean Theorem: Định lý Pitago

opposite: Đối diện

adjacent: Kề, kề nhau

base: Cạnh đáy

height: Chiều cao

perpendicular: Vuông góc

area: Diện tích

perimeter: Chu vi.

perpendicular side or perpendicular leg: Cạnh góc vuông

respectively: Tương ứng, theo thứ tự.

solution: Giải, lời giải, nghiệm (đối với phương trình).

certain: Cho trước, đã biết, chắc chắn.

Certain sets of integers: Tập hợp các số nguyên cho trước. 

A triangle is a closed figure with three sides, each side being a line segment.
The sum of the angles of a triangle is always 180°.

Scalene triangles are triangles with no two sides equal. Scalene triangles also have no two angles equal.

Properties-of-triagles-Part-1_Scalene triangle

Isosceles triangles have two equal sides and two equal angles formed by the equal sides and the unequal side. See the figure below.


a = b

A = B

C = 180° – 2(A)

Equilateral triangles have all three sides and all three angles equal. Since the sum of the three angles of a triangle is 180°, each angle of an equilateral triangle is 60°.


a = b = c

∠A = ∠B = ∠C = 60°

A right triangle has one angle equal to a right angle (90°). The sum of the other two angles of a right triangle is, therefore, 90°. The most important relationship in a right triangle is the Pythagorean Theorem. It states that c2 = a2 + b2, where c, the hypotenuse, is the length of the side opposite the right angle, and a and b are the lengths of the other two sides.


Example: If the two sides of a right triangle adjacent to the right angle are 3 inches and 4 inches respectively, find the length of the side opposite the right angle.


Use the Pythagorean Theorem, c2 = a2 + b2, where a = 3 and b = 4. Then, c2 = 32 + 42 or c2 = 9 + 16 = 25. Thus c = 5.



———End of properties of triangle part 1———

English for math – Vocabulary of Fractions



Fraction: Phân số

Numerator: Tử số

Denominator: Mẫu số

In the fraction 8/13, the numerator is 8 and the denominator is 13.

Equivalent fractions: Phân số tương đương (2/3; 4/6;…)

The lowest common denominator: Mẫu số chung nhỏ nhất

Common factor: Nhân tử chung

Add: Cộng

Minus: Trừ

Multiply: Nhân

Divide: Chia; divisible: Có thể chia được; Có thể chia hết.

Times: Nhân; lần

Equal to; Equals; is: Bằng

Decimal: Số thập phân

Percent: Phần trăm

Sign: Ký hiệu

Odd: Số lẻ

Even: Số chẵn

Largest number: Số lớn nhất

Smallest number: Số nhỏ nhất

Obtain: Đạt được; thu được

Convert: Chuyển

Reduce: Đơn giản; giản lược

Drop: Loại bỏ, xóa bỏ

Carry out: Thực hiện; Tính toán

Procedure: Thủ tục (thường được sử dụng trong chương trình, còn gọi là chương trình con, tương tự như Funtion).


Change the fraction

– To change a fraction to a decimal, divide the numerator of the fraction by its denominator.


Express 5/6 as a decimal. We divide 5 by 6, obtaining 0.83.

– To convert a decimal to a fraction, delete the decimal point and divide by whatever unit of 10 the number of decimal places represents.


Convert 0.83 to a fraction.

+ First, delete the decimal point.

+ Second, two decimal places represent hundredths, so divide 83 by 100: 83/100
0.83 = 83/100

– To change a fraction to a percent, find its decimal form, multiply by 100, and add a percent sign.


Express 3/8 as a percent.

To convert 3/8 to a decimal, divide 3 by 8, which gives us 0.375. Multiplying 0.375 by 100 gives us 37.5%.

– To change a percent to a fraction, drop the percent sign and divide the number by 100.


Express 17% as a fraction.

Dropping the % sign gives us 17, and dividing by 100 gives us 17/100

– To reduce a fraction, divide the numerator and denominator by the largest number that divides them both evenly.

Example 1:

Reduce 10/15 .

Dividing both the numerator and denominator by 5 gives us 2/3

Example 2:

Reduce 12/36

The largest number that divides into both 12 and 36 is 12.

Reducing the fraction, we have 1/3.

Note: In both examples, the reduced fraction is exactly equal to the original fraction: 2/3 = 10/15 and 12/36 = 1/3

Add fractions

– To add fractions with like denominators, add the numerators of the fractions, keeping the same denominator.

Example: 1/7 + 2/7 + 3/7 = 6/7

– To add fractions with different denominators, you must first change all of the fractions to equivalent fractions with the same denominators.

STEP 1. Find the lowest (or least) common denominator, the smallest number divisible by all of the denominators.


If the fractions to be added are 1/3, 1/4 and 5/6, then the lowest common denominator is 12, because 12 is the smallest number that is divisible by 3, 4, and 6.

STEP 2. Convert all of the fractions to equivalent fractions, each having the lowest common denominator as its denominator.

To do this, multiply the numerator of each fraction by the number of times that its denominator goes into the lowest common denominator.

The product of this multiplication will be the new numerator.

The denominator of the equivalent fractions will be the lowest common denominator. (See Step 1 above.)


The lowest common denominator of are 1/3, 1/4 and 5/6 is 12.

Thus, 1/3 = 4/12 because 12 divided by 3 is 4, and 4 times 1 =  4.  1/4 = 3/12, because 12 divided by 4 is 3, and 3 times 1 = 3. 5/6 = 10/12, because 12 divided by 6 is 2, and 2 times 5 = 10.

STEP 3. Now add all of the equivalent fractions by adding the numerators.


4/12 + 3/12 + 10/12 = 17/12

STEP 4. Reduce the fraction if possible.


Add 4/5, 2/3, and 8/15.

The lowest common denominator is 15, because 15 is the smallest number that is divisible by 5, 3, and 15.

Then, 4/5 is equivalent to 12/15; 2/3 is equivalent to 10/15; and 8/15remains as 8/15.

Adding these numbers gives us 12/15 + 10/15 + 8/15 = 30/15.

Both 30 and 15 are divisible by 15, giving us 2/1, or 2.

Multiply fractions

To multiply fractions, follow this procedure:

STEP 1. To find the numerator of the product, multiply all the numerators of the fractions being multiplied.

STEP 2. To find the denominator of the product, multiply all of the denominators of the fractions being multiplied.

STEP 3. Reduce the product.


5/7 x 2/15 = 1/7 x 2/3 = 2/21.

We reduced by dividing both the numerator and denominator by 5, the common factor.

Divide fractions

To divide fractions, follow this procedure:

STEP 1. Invert the divisor. That is, switch the positions of the numerator and denominator in the fraction you are dividing by.

STEP 2. Replace the division sign with a multiplication sign.

STEP 3. Carry out the multiplication indicated.

STEP 4. Reduce the product.


Find 3/4 : 7/8.

Inverting 7/8, the divisor, gives us 8/7.

Replacing the division sign with a multiplication sign gives us 3/4 x 8/7. Carrying out the multiplication gives us 3/4 x 8/7 = 24/28.

The fraction 24/28 may then be reduced to 6/7 by dividing both the numerator and the denominator by 4.

Tải bài viết

The name of all Triangles

Name of triangles - The-Triangle-Test

A triangle is a geometrical figure bounded by three straight lines and having three angles. Such a definition may be correct, but it gives one the idea that a triangle is a decidedly uninteresting figure. There are many different kinds of triangles and each one has its own interesting peculiarities.

Bài viết liên quan:

From the information given, can you state the names of these triangles?

1. I am readily suggested when you look at a trillium.

2. I appear when a man stands on level ground with his legs straight and his feet slightly apart.

3. I have a special name derived from a Greek word meaning “uneven.”

4. I am formed by joining the feet of the perpendiculars from the vertices of a triangle to the opposite sides.

5. The sum of the squares on two of my sides equals the square constructed on my third side.

6. There are at least two of us. We find that our corresponding angles are equal and our sides are proportional.

7. The sides and the diagonals of a quadrilateral are used to construct me.

8. My sides are not straight lines and the sum of my angles is greater than 1800.

9. I have gained the title “pons asinorum” for a certain proposition in Euclid.

10. I am connected with the stars and the zenith.

New vocabulary

geometric: Hình học; geometrical: thuộc về hình học.

bound: Giới hạn, ranh giới, bao bởi.

figure: (toán học) Hình, con số.

peculiarity: Đặc thù, tính chất riêng.

slightly: Nhỏ, mỏng mảnh, yếu ớt.

uneven: (toán học) Lẻ (= odd).

derive: Xuất phát từ, bắt nguồn từ, lấy được từ,…

perpendicular: Góc vuông, đường vuông góc, mặt phẳng vuông góc, trực giao, thẳng đứng, thẳng góc.

vertices, số nhiều của vertex: (toán học) Đỉnh.

proportional: (toán học): Số hạng của tỷ lệ thức.

diagonal: Đường chéo, chéo.

quadrilateral: Hình bốn cạnh, hình tứ giác.

angle: Góc

pons asinorum: Theo wiki “In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum“.

proposition: (toán học) Sự trình bày một định lý hoặc vấn đề (có chứng minh); Định đề.

zenith: Điểm cao nhất, cực điểm.

 Answer key

As the name suggests, all the sides are equal. Equilateral triangles have not only equal sides but have equal angles too and can thus be called equiangular triangles. The trillium has three axes of symmetry suggesting an equilateral triangle. Many plants have geometrical shapes in their roots, stems, leaves, and flowers. What shape is found in the cross-section of the stem of the sedge family?


The word means “equal legs.” Any triangle which has two of its three sides equal is called an isosceles triangle. If we look around us we can often find geometrical figures in nature. In the construction of houses, ships, and aircraft, the isosceles triangle is often encountered.

SCALENE (Thường)

This is a triangle which is “uneven” because it has all its sides unequal. The word “scalene” has nothing to do with drawing to scale, but is derived from the Greek word “skalenos,” which means “uneven” or “unequal.”

4. PEDAL (Thủy tức hay Thùy tức)

This triangle is sometimes called the orthocentric triangle. If AD, BE, and CF are the perpendiculars dropped from the vertices of the triangle ABC to the opposite sides, then the
triangle DEF is the pedal triangle. The three perpendiculars pass through a common point called the orthocenter.

Tham khảo thêm


The unique property of the right-angled triangle ABC is that if angle A is the right angle then a2 = b2 + c2. The geometrical proof of this property is associated with the Greek mathematician Pythagoras, who lived in the sixth century B.C. Triangles whose sides are in the ratios 3-4-5 or 5-12-13 are right-angled.

6. SIMILAR (Đồng dạng)

Two triangles are said to be similar if they have their angles equal to one another and have their sides, taken in order, about the corresponding equal angles proportional. The areas of similar triangles are proportional to the squares of their corresponding sides.

7. HARMONIC (Điều hoà)

ABCD is the quadrilateral and OXY is the harmonic triangle. It is formed by joining the intersection points of the sides and the intersection point of the diagonals.

Name of Triangles - Harmonic-Triangle

Tham khảo thêm


A spherical triangle consists of a portion of a sphere bounded by three arcs of great circles. Obviously the sides do not consist of straight lines. The sum of the three angles of such a triangle lies between 1800 and 5400. Much of the early work with spherical triangles was done by Menelaus about A.D. 100.

Tham khảo thêm


The particular proposition proved that the angles at the base of an isosceles triangle are equal. The title “pons asinorum” means the “bridge of asses.” It is said that in the Middle Ages the “donkey” could not pass over this bridge to continue his study of Euclidean geometry, but the name may be due to the fact that the figure in Euclid resembles a simple truss bridge.


This is a spherical triangle on a celestial sphere which has for its vertices the nearest celestial pole, the zenith, and the star under consideration.

Tham khảo thêm

Comparison symbols vocabulary for math

Math Comparison symbols vocabulary


100: One hundred

101: One hundred and one

110: One hundred and ten

115: One hundred and fifteen

200: Two hundred

300: Three hundred

400: Four hundred

500: Five hundred

600: Six hundred

700: Seven hundred

800: Eight hundred

900: Nine hundred

1000: One thousand

Math Comparison Symbols

=       Equals : Bằng

1+3 = 4

One plus three equals / is / equal to four.

≠        Not equal to : Không bằng

2×3 ≠ 5

2 multiply 3 is not equal to 5

>       Greater than : Lớn hơn


Fifteen is greater than six plus 7

≥        Greater than or equal to : Lớn hơn hoặc bằng


Eighteen is greater than or equal to 2 multiply 9.

<       Less than : Nhỏ hơn


Twenty is less than thirty.

≤        Less than or equal to : Nhỏ hơn hoặc bằng

10-3 ≤ 20-3

Teen minus three is less than or equal to twenty minus three.

Classifying numbers

Negative: Số âm.

Là các số có dấu trừ (-) phía trước.

Ví dụ: -4, -120,…

Positive: Số dương.

Là các số có dấu cộng (+) phía trước hoặc không dấu.

Ví dụ: 4, +120,…

Odd: Số lẻ.

Là các số 1, 3, 5, 7,…

Even: Số chẵn.

Là các số 2,4,6,8,…

Integer: Số nguyên (hoặc whole number).

Số nguyên là tập hợp bao gồm các số không, số tự nhiên dương và các số đối của chúng còn gọi là số tự nhiên âm.

Real: Số thực.

Số thực là tập hợp bao gồm số dương(1,2,3), số 0, số âm(-1,-2,-3), số hữu tỉ (42, -23/45), số vô tỉ (số pi, số √2).

Nói một cách đơn giản hơn thì số thực là tập hợp các số hữu tỉ và vô tỉ.

Decimal: Số lẻ (dấu chấm (.) được gọi là decimal point).

Rational Numbers: Số hữu tỉ

Trong toán học, số hữu tỉ là các số x có thể biểu diễn dưới dạng phân số (thương) a/b, trong đó a và b là các số nguyên với b 0.

Irrational numbers: Số vô tỷ

Số vô tỉ là tập hợp các số viết được dưới dạng số thập phân vô hạn không tuần hoàn.

6 trang web học tiếng Anh miễn phí hiệu quả nhất

Các trang web học tiếng Anh tin cậy hiệu quả miễn phí

Bạn học tiếng Anh bằng cách nào? Sử dụng thời gian rảnh của bạn ra sao? Mùa hè đến rồi, mình giới thiệu bạn 6 trang web học tiếng Anh miễn phí khá tốt nhưng đáng tin cậy mà mình đã chọn lọc.

1. Học tiếng Anh với BBC

BBC (viết tắt British Broadcasting Corporation, theo, là thông tấn xã quốc gia của Vương quốc Liên hiệp Anh và Bắc Ireland. BBC là đài truyền hình quốc gia lâu đời nhất trên thế giới cũng như là đài truyền hình lớn nhất thế giới theo số lượng nhân viên.

Trang giảng dạy tiếng Anh đa dạng, đủ mọi cấp học (ngoại trừ mẫu giáo hoặc chưa biết gì về tiếng Anh).

  Học tiếng Anh -BBC

Trang web này giúp bạn học tiếng Anh hiệu quả:

– Bạn không cần đăng ký (Register) như các trang web khác.

– Phần nghe có Transcript giúp bạn theo dõi những đoạn khó nghe.

– Ngoài ra, có mục Business (English at Work), trình độ trung bình (intermediate) để bạn học những tình huống thực tế.

2. Học tiếng Anh với BritishCouncil

BritishCouncil (Hội đồng Anh) nổi tiếng là tổ chức thi để cấp chứng chỉ quốc tế trên toàn thế giới, điển hình là chứng chỉ IELTS (International English Language Testing System)

Học tiếng Anh -Britishcouncil

Bạn có thể tìm hiểu các đề mục và kỹ năng để học, rất phong phú. Theo mình đánh giá, trang web này đa dạng hơn trang, nếu bạn muốn học tiếng Anh online thì đây là trang web nên xem xét.

– Các video đều có Transcript (giống

– Sau mỗi video đều có câu hỏi để bạn kiểm chứng những gì đã nghe, thấy.

– Mục tiếng Anh thương mại (Business English) khá phong phú.

– Ngoài ra, còn có mục TeachingEnglish để các bạn giảng dạng tiếng Anh tham khảo.

3. Học tiếng Anh với

Mới nghe qua tên, chắc bạn đã hình dung được trang web này đề cập đến phần nào trong tiếng Anh? Những trang web giảng dạy ngữ pháp (grammar) nhiều lắm, bạn muốn học ở đâu cũng được. Tuy nhiên, nếu chưa có nguồn nào để học thì bạn nên xem xét trang web này.

Học tiếng Anh - Englishgrammar

Mặc dù ngữ pháp không phải là một kỹ năng riêng để đánh giá qua các kỳ thi quốc tế, nhưng bạn chưa vững ngữ pháp sẽ ảnh hưởng đến một số kỹ năng khác rất nhiều.

4. Học tiếng Anh với

Ngược với ngữ pháp, ồn ào nhất trong việc học tiếng Anh là đây Đúng với tên “basic”, trang này gợi ý mẫu câu, từ vựng, file PDF,… trình độ trung bình trở lại rất nhiều các chủ đề thường gặp trong cuộc sống.

Học tiếng Anh - BasicEnglishSpeaking

Đáp ứng tốt cho những bạn có ý định dự thi A2, B1 khung 6 bậc của Việt nam chúng ta.

5. Học tiếng Anh với

Trích khẩu hiểu của

“Duolingo được sử dụng phổ biến, từ người giàu nhất thế giới, các diễn viên Hollywood đến những em học sinh tại trường công ở các nước đang phát triển. Chúng tôi tin vào quyền bình đẳng, mọi người đều xứng đáng có được một nền giáo dục chất lượng cao dù họ giàu hay nghèo.”

Học tiếng Anh -Duolingo

Nếu bạn cần hình ảnh, màu sắc thì trang web này giành cho bạn, tập trung vào trẻ nhỏ (kids) nên giao diện rất đẹp mắt.

* Hỗ trợ rất nhiều ngôn ngữ hiển thị, bạn có thể học song ngữ Anh – Việt hay Anh – Hàn Quốc,… đều được.

* Bài tập để tự luyện rất nhiều (nhưng để khai thác hết, phải tốn phí nghe bạn?).

6. Học tiếng Anh với

Nhìn thấy tên trang web không nói lên được điều gì các bạn, nhưng đây là một trang chuyên về giáo dục phổ thông (Maths và English).

Học tiếng Anh -IXL

– Giao diện đẹp (chủ yếu dành cho học sinh).

– Mục MathsEnglish từ Reception đến Year 13 (có lẽ từ mẫu giáo đến học sinh năm thứ 13, thấy cần thiết thì các bạn tìm hiểu thêm).

– Riêng mục Maths mình thấy rất hay, nếu bạn có định hướng học từ vựng Toán tiếng Anh (English for maths) thì nên xem xét trang này. Mình nhận thấy trang này thiết kế theo kiểu chơi mà học (play school) miễn phí rất tốt. Các bạn có thể từng bước học từ vựng toán học tiếng Anh luôn.

Kết luận

Học tiếng Anh là một quá trình. Nếu bạn học với thầy cô, có trường có lớp hẳn hoi là một giải pháp không thể thay thế. Nếu bạn tự học thì nên tìm hiểu những trang web tin cậy (đặc biệt là phần nghe, nói phải giọng người bản xứ).

Chúc bạn học tiếng Anh mau đạt kết quả.

Practice shapes vocabulary for math

I. Count number of triangles    

The classic example is that of the triangular “matchstick” configuration as shown below.

The usual first step is simply to enumerate all of the distinct shape types of interest. Oftentimes, finding all of the shape types is the most tricky and potentially treacherous step—it can take some experience and meticulousness not to miss anything. Luckily, for this problem, it is relatively simple to enumerate all of the cases. Clearly, there are only three possible sizes of triangles: those having side length 1, side length 2, and side length 3.

The next step is to take a “census” of the different shape types. In this problem, it is fairly straightforward to do the census directly.

There are triangles of side length 1:

There are  triangles of side length 2:

Finally, there is only  triangle of side length 3:

Thus, in total, there are triangles.

Post in the same topic: Shapes vocabulary for math

II. Fill information

For each of the shapes below,

* Shade the parallel lines, using a different color for each pair;

* Write down the number of pairs of parallel lines;

* Mark in the right angles using a square. The first one is done for you.

Number 1

* Slides: 4

* Right angles: 4

* Pairs of parallel lines: 2

Number 2-6?

 III. Identifying shapes

Name of shape: Rectangle, Triangle, Square, Circle, Parallelogram, Trapezoid (hoặc trapezium)

Sides: Rỗng, 3, 4.

Corners: Rỗng, 3, 4

Need to know:

+ All sides are the same length. The boxes on a checkerboard are square.

+ Yield signs are triangles, and a piece of bread can easily be cut into two triangles.

+ Stoplights are circles, and so are most dinner plates oval   It looks like a squashed circle.

+ It has two pairs of sides that are the same length. A ping-pong table is a rectangle.

+ Only one pair of sides is parallel, and the sides that are parallel aren’t the same length. If you cut the top off of a pyramid, you would have a trapezoid.

+ It’s like a rectangle that leaned over.

The rectangle is done for you.

Name of shape: Rectangle

Sides: 4

Corners: 4

Need to know: It has two pairs of sides that are the same length. A ping-pong table is a rectangle.

Answer key:

Need to know, Corners, Sides, Name of shape:

– Yield signs are triangles, and a piece of bread can easily be cut into two triangles; 3; 3; Triangle.

– All sides are the same length. The boxes on a checkerboard are square; 4; 4; Square.

–  Stoplights are circles, and so are most dinner plates oval. It looks like a squashed circle; Rỗng; Rỗng; Circle.

– It’s like a rectangle that leaned over; 4;4; Parallelogram.

– Only one pair of sides are parallel, and the sides that are parallel aren’t the same length. If you cut the top off of a pyramid, you would have a trapezoid; 4;4; Trapezoid (hoặc trapezium).