Math Vocab
Corresponding Angles
When two lines are crossed by another line (called the Transversal): The angles in matching corners are called Corresponding Angles. In this example, these are corresponding angles: a and e b and f c and g d and h
Septagon
A 7-sided polygon (a flat shape with straight sides).
Coefficient
A Coefficient is a number used to multiply a variable (4x means 4 times x, so 4 is a coefficient)
Variable
A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y.
Circle
A circle is easy to make: Draw a curve that is "radius" away from a central point. And so: All points are the same distance from the center. The circle is a plane shape (two dimensional): And the definition of a circle is: The set of all points on a plane that are a fixed distance from a center.
Decagon
A decagon is a polygon with ten sides. Find the area of the regular decagon with sides of length 14.4 cm and apothem 17.3 cm. Choices: A. 1255.60 B. 622.80 C. 1245.60 D. 830.40 Correct Answer: C Solution: Step 1: Perimeter P of the regular decagon = n × s = 10 × 14.4 = 144 cm. Step 2: Area A of the regular decagon ap/2 = (17.3x144)/2
Hexagon
A hexagon is a 6-sided polygon (a flat shape with straight sides). Regular or Irregular If all angles are equal and all sides are equal, then it is regular, otherwise it is irregular: Regular Hexagon Irregular Hexagons Concave or Convex A convex hexagon has no angles pointing inwards. More precisely, no internal angles can be more than 180°. If there are any internal angles greater than 180° then it is concave. (Think: concave has a "cave" in it)
Definition of Division
A mathematical operation that indicates how many equal quantities add up to a specific number. 8 ÷ 4 = 2 is an example of division.
Definition of Subtraction
A mathematical operation that takes away a quantity from a larger whole. 4 - 2 = 2 is an example of subtraction.
Pentagon
A pentagon is a 5-sided Polygon (a flat shape with straight sides): If all angles are equal and all sides are equal, then it is regular, otherwise it is irregular: Regular Pentagon Irregular Pentagons Concave or Convex A convex pentagon has no angles pointing inwards. More precisely, no internal angles can be more than 180°. If there are any internal angles greater than 180° then it is concave. (Think: concave has a "cave" in it)
Vertex
A point at which the arms (sides) of an angle meet is known as the vertex of the angle. More about Vertex of an Angle Measuring of angle is done about the vertex.
Plane
A point exists in zero dimensions. A line exists in one dimension, and we specify a line with two points. A plane exists in two dimensions. We specify a plane with THREE points. Any two of the points specify a line. All possible lines that pass through the third point and any point in the line make up a plane. In more obvious language, a plane is a flat surface that extends indefinitely in its two dimensions, length and width. A plane has no height.
Point
A point is an exact location in space. Points are dimensionless. That is, a point has no width, length, or height. We locate points relative to some arbitrary standard point, often called the "origin". Many physical objects suggest the idea of a point. Examples include the tip of a pencil, the corner of a cube, or a dot on a sheet of paper.
Polygon
A polygon is a plane shape with straight sides. Is it a Polygon? Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).
Proportion
A proportion is a name we give to a statement that two ratios are equal. It can be written in two ways: two equal fractions, or, using a colon, a:b = c:d When two ratios are equal, then the cross products of the ratios are equal. That is, for the proportion, a:b = c:d , a x d = b x c Ratios are useful ways to compare two quantities. But how do you compare ratios? For example, in Figure 1 below two out of the three circles are shaded, and in Figure 2 below four out of the six circles are shaded. Although Figure 2 has more circles, the ratio of shaded circles to total circles is the same. That is,. A statement such as this that one ratio is equal to another is called a proportion. Proportional reasoning involves the ability to compare and produce equal ratios. A common use of proportions occurs when making or using maps and scale models. There are several ways to solve a proportion. One is related to how you find equivalent fractions. To find equivalent fractions, you multiply or divide the numerator and denominator by the same number. Thus, to solve , note that the numerator (2) was multiplied by 3 in order to get 6. Then do the same to the denominator (3) to get n=9. This works well when the numerator and denominator of one fraction are multiples of the other fraction, but it is more difficult to do when they are not, as in the case of . In this case we multiply both sides of the equation by 8n as shown below: (multiplying both sides by 8n) (divide out the common factors) (divide both sides by 6) (simplify)
Pyramid
A pyramid is made by connecting a base to an apex The Volume of a Pyramid 1/3 × [Base Area] × Height The Surface Area of a Pyramid When all side faces are the same: [Base Area] + 1/2 × Perimeter × [Slant Length] When side faces are different: [Base Area] + [Lateral Area]
Ratio
A ratio says how much of one thing there is compared to another thing. There are 3 blue squares to 1 yellow square Ratios can be shown in different ways: Using the ":" to separate the values: 3 : 1 Instead of the ":" you can use the word "to": 3 to 1 Or write it like a fraction: 3 1 A ratio can be scaled up: Here the ratio is also 3 blue squares to 1 yellow square, even though there are more squares. Using Ratios The trick with ratios is to always multiply or divide the numbers by the same value. Example: 4 : 5 is the same as 4×2 : 5×2 = 8 : 10 Recipes Example: A Recipe for pancakes uses 3 cups of flour and 2 cups of milk. So the ratio of flour to milk is 3 : 2 If you needed to make pancakes for a LOT of people you might need 4 times the quantity, so you multiply the numbers by 4: 3×4 : 2×4 = 12 : 8
Ray
A ray is a part of a line that begins at a particular point (called the endpoint) and extends endlessly in one direction. A ray is also called half-line. How do we name a Ray? A ray is named based on the direction in which it extends. A ray is named with its endpoint in the first place, followed by the direction in which its moving. In the example shown below, P is the endpoint and Q is the point towards which the ray extends. So, the ray PQ is represented as: Look at another example. This ray will be called QP as it starts at Q and extends towards P. So, the ray QP is represented as: Solved Example on Ray Which of the following is a ray? Solution: Step 1: The definition of a ray states that "A ray is a part of a line that begins at a particular point (called the endpoint) and extends endlessly in one direction." Step 2: As per the defition of a ray, among the figures given only Figure 2 represents a ray.
Prism
A solid object that has two identical ends and all flat sides. The cross section is the same all along its length. The shape of the ends give the prism a name, such as "triangular prism" It is also a polyhedron.
Cylinder
A solid object with: * two identical flat ends that are circular or elliptical * and one curved side. It has the same cross-section from one end to the other.
Cone
A solid that has a circular base and a single vertex.
Straight Angles
A straight angle is 180 degrees This is a straight angle A straight angle changes the direction to point the opposite way. Sometimes people say "You did a complete 180 on that!" ... meaning you completely changed your mind, idea or direction.
Diagonal
A straight line inside a shape that goes from one corner to another (but not an edge). So, if you join two corners (called "vertices") of a polygon which are not already joined by an edge, you get a diagonal.
Trapezoid
A trapezoid is a 4-sided flat shape with straight sides that has a pair of opposite sides parallel. The parallel sides are the "bases" The other two sides are the "legs" The distance (at right angles) from one base to the other is called the "altitude" The Area is the average of the two base lengths times the altitude Area = (6 m + 4 m)/2 × 3 m = 5 m × 3 m = 15 m2 The Perimeter is the distance around the edges. Perimeter = 5 cm + 12 cm + 4 cm + 15 cm = 36 cm
Triangle
A triangle has three sides and three angles The three angles always add to 180°
Acute Triangle
A triangle that has all angles less than 90°
Acute Angles
Acute Triangle is a triangle with its angles less than 90 degrees. Example of An Acute Triangle The triangle shown has all its angles measuring less than 90 degrees.
Addition Property of Equality
Addition Property of Equality: If the same number is added to both sides of an equation, the two sides remain equal. That is, if x = y, then x + z = y + z.
Obtuse Triangle
All angles are less than 90°
Zero Property of Multiplication
Also called the zero product property. There exists a unique number, zero, such that the product of any Real number x and 0 is always equal to 0
Expression
An Expression is a group of terms (the terms are separated by + or - signs)
Angle
An angle is formed by two rays with a common endpoint (called the vertex). Examples of An Angle In the figure shown, angle AOB is formed by the rays OA and OB with a common endpoint O.
Equation
An equation says that two things are equal. It will have an equals sign "=" like this: x + 2 = 6 That equations says: what is on the left (x + 2) is equal to what is on the right (6) So an equation is like a statement "this equals that"
Octagon
An octagon is an 8-sided polygon (a flat shape with straight sides).
Obtuse Angles
Any angle greater than 90° and smaller than 180° is called an Obtuse Angle. More about Obtuse Angle An angle of 90° is called a right angle. An angle of 180° is called a straight angle.
Area of a square:
Area = a² = 1² = 1
Area of a rectangle
Area = w × h =3 × 1 = 3
Area of a Triangle formula:
Area = ½b × h = ½ × 1 × 1 = 0.5
Area of a Triangle at 60 degree angle:
Area = ½r²θ = ½ × 1² × 60 = 30
Area of a Circle formula:
Area = π × r² = π × 1² = 3.141592653589
Right Angles
Right angle is an angle that has a measure of 90°. Examples of Right Angle The figure below shows a right angle.
Line
As for a line segment, we specify a line with two points. Starting with the corresponding line segment, we find other line segments that share at least two points with the original line segment. In this way we extend the original line segment indefinitely. The set of all possible line segments findable in this way constitutes a line. A line extends indefinitely in a single dimension. Its length, having no limit, is infinite. Like the line segments that constitute it, it has no width or height. You may specify a line by specifying any two points within the line. For any two points, only one line passes through both points. On the other hand, an unlimited number of lines pass through any single point.
Associative Property of Addition/Multiplication
Associative Property The word "associative" comes from "associate" or "group";the Associative Property is the rule that refers to grouping. For addition, the rule is "a + (b + c) = (a + b) + c"; in numbers, this means 2 + (3 + 4) = (2 + 3) + 4. For multiplication, the rule is "a(bc) = (ab)c"; in numbers, this means 2(3×4) = (2×3)4. Any time they refer to the Associative Property, they want you to regroup things; any time a computation depends on things being regrouped, they want you to say that the computation uses the Associative Property. Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved Rearrange, using the Associative Property: 2(3x) They want you to regroup things, not simplify things. In other words, they do not want you to say "6x". They want to see the following regrouping: (2×3)x Simplify 2(3x), and justify your steps. In this case, they do want you to simplify, but you have to tell why it's okay to do... just exactly what you've always done. Here's how this works: 2(3x) original (given) statement (2×3)x by the Associative Property 6x simplification (2×3 = 6) Why is it true that 2(3x) = (2×3)x? Since all they did was regroup things, this is true by the Associative Property.
Commutative Property of Addition/Multiplication
Commutative Property The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is "ab = ba"; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property. Use the Commutative Property to restate "3×4×x" in at least two ways. They want you to move stuff around, not simplify. In other words, the answer is not "12x"; the answer is any two of the following: 4 × 3 × x, 4 × x × 3, 3 × x × 4, x × 3 × 4, and x × 4 × 3 Why is it true that 3(4x) = (4x)(3)? Since all they did was move stuff around (they didn't regroup), this is true by the Commutative Property.
Congruent Angles
Congruent means having the same measure. More about Congruent Congruent angles are angles that have the same measure. An equilateral triangle has all its angles measuring 60°. So, the angles of an equilateral triangle are congruent. A square and a rectangle have all their angles measuring 90°. So, the angles of a square and a rectangle are congruent. Congruent Figures have the same shape and size. Congruent Segments have the same length.
Same Side Interior Angles
Created where a transversal crosses two (usually parallel) lines. Each pair of interior angles are inside the parallel lines, and on the same side of the transversal.
Degree
Degree is the basic unit for measuring the size of an angle. It is equivalent to of a full rotation or full angle. More about Degree (Angle) Degrees are represented by the symbol ' ° '. A protractor is used to measure the angles in degrees.
Distributive Property of Multiplication
Distributive Property The Distributive Property is easy to remember, if you recall that "multiplication distributes over addition". Formally, they write this property as "a(b + c) = ab + ac". In numbers, this means, that 2(3 + 4) = 2×3 + 2×4. Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or factor something out); any time a computation depends on multiplying through a parentheses (or factoring something out), they want you to say that the computation used the Distributive Property. Why is the following true? 2(x + y) = 2x + 2y Since they distributed through the parentheses, this is true by the Distributive Property. Use the Distributive Property to rearrange: 4x - 8 The Distributive Property either takes something through a parentheses or else factors something out. Since there aren't any parentheses to go into, you must need to factor out of. Then the answer is "By the Distributive Property, 4x - 8 = 4(x - 2)" "But wait!" you say. "The Distributive Property says multiplication distributes over addition, not subtraction! What gives?" You make a good point. This is one of those times when it's best to be flexible. You can either view the contents of the parentheses as the subtraction of a positive number ("x - 2") or else as the addition of a negative number ("x + (-2)"). In the latter case, it's easy to see that the Distributive Property applies, because you're still adding; you're just adding a negative. The other two properties come in two versions each: one for addition and the other for multiplication. (Note that the Distributive Property refers to both addition and multiplication, too, but to both within just one rule.)
Division Property of Equality
Division Property of Equality states that dividing both sides of an equation by a non-zero number doesn't affect the equation. Examples of Division Property of Equality 5p = 15 By dividing both sides of the equation by 5, we get p = 3. Substituting p = 3 in the original equation will help us check the division property of equality. 5 × 3 = 15, it's true! Solved Example on Division Property of Equality Solve - 3x = 12 using division property of equality. Choices: A. - 4 B. 3 C. 12 D. 36 Correct Answer: A Solution: Step 1: - 3x = 12 Step 2: [Apply division property of equality.]
Base
Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 53. The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The thing that's being multiplied, being 5 in this example, is called the "base".
Power
Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 53. The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The thing that's being multiplied, being 5 in this example, is called the "base". This process of using exponents is called "raising to a power", where the exponent is the "power". The expression "53" is pronounced as "five, raised to the third power" or "five to the third". There are two specially-named powers: "to the second power" is generally pronounced as "squared", and "to the third power" is generally pronounced as "cubed". So "53" is commonly pronounced as "five cubed".
Quadrilateral
Four sides (edges) Four vertices (corners) The interior angles add up to 360 degrees:
Right Triangle
Has a right angle (90°)
Volume
Here, we provide you with volume formulas for some common three-dimensional figure Cube: Volume = a3 = a × a × a Cylinder: Volume = pi × r2 × h pi = 3.14 h is the height r is the radius Rectangular solid: Volume = l × w × h l is the length w is the width h is the height Sphere: Volume = (4 × pi × r3)/3 pi = 3.14 r is the radius Cone: Volume = (pi × r2 × h)/3 pi = 3.14 r is the radius h is the height Pyramid: Volume = (B × h)/3 B is the area of the base h is the height
Identity Property of Addition
Identity property of addition states that the sum of zero and any number or variable is the number or variable itself. For example, 4 + 0 = 4, - 11 + 0 = - 11, y + 0 = y are few examples illustrating the identity property of addition.
Identity Property of Multiplication
Identity property of multiplication states that the product of 1 and any number or variable is the number or variable itself. For example, 4 × 1 = 4, - 11 × 1 = - 11, y × 1 = y are few examples illustrating the identity property of multiplication.
Regular Polygon
If all angles are equal and all sides are equal, then it is a regular polygon (otherwise it is "irregular"). This is a regular pentagon.
Rhombus
If all sides of parallelogram are equal, then this parallelogram is called a rhombus ( Fig.34 ) . Diagonals of a rhombus are mutually perpendicular ( AC BD ) and divide its angles into two ( DCA = BCA, ABD = CBD etc. ).
Rectangle
If one of angles of parallelogram is right, then all angles are right (why ?). This parallelogram is called a rectangle ( Fig.33 ). Main properties of a rectangle. Sides of rectangle are its heights simultaneously. Diagonals of a rectangle are equal: AC = BD. A square of a diagonal length is equal to a sum of squares of its sides' lengths ( see above Pythagorean theorem ): AC² = AD² + DC².
Reciprocal
If the product of two numbers is 1, then the two numbers are said to be reciprocals of each other. In other words, a reciprocal is the multiplicative inverse of a number. The reciprocal of 'a' is . Examples of Reciprocal Consider . 9 is the reciprocal of is the reciprocal of 9.
Substitute
In Algebra "Substitution" means putting numbers where the letters are: Example 1: If x=5 then what is 10/x + 4 ? Put "5" where "x" is: 10/5 + 4 = 2 + 4 = 6
Face and Edge
In any geometric solid that is composed of flat surfaces, each flat surface is called a face. The line where two faces meet is called an edge. For example, the cube above has six faces, each of which is a square. Where two squares meet, a line segment is formed, which is called an edge. In the case of a cube, it has 12 such edges.
Factor
In the problem 3 x 4 = 12, 3 and 4 are factors and 12 is the product. Factoring is like taking a number apart. It means to express a number as the product of its factors. Factors are either composite numbers or prime numbers (except that 0 and 1 are neither prime nor composite). The number 12 is a multiple of 3, because it can be divided evenly by 3. 3 x 4 = 12 3 and 4 are both factors of 12 12 is a multiple of both 3 and 4.
Segment
Line segment is the part of a line consisting of two endpoints and all points between them. Examples of Line Segment In the figure shown, PQ, PR, and RQ are line segments. Solved Example on Line Segment Which of the following can be obtained by joining line segments? Choices: A. a circle B. a polygon C. a line D. a sphere Correct Answer: B Solution: Step 1: By joining line segments, we can form a polygon.
Multiplication Property of Equality
Multiplicative Property of Equality: The two sides of an equation remain equal if they are multiplied by the same number. That is: for any real numbers a, b, and c, if a = b, then ac = bc. For example: Multiplicative Property of Equality: x = 4 ⇒ 3x = 12
Multiplication Property of -1
Multiplicative property of Negative One: The product of - 1 and any number results in the opposite of that number. That is, for any real number a, a × - 1 = - a.
Scalene Triangle
No equal sides No equal angles
Parallel Lines
Parallel Lines are distinct lines lying in the same plane and they never intersect each other. Parallel lines have the same slope. In the figure below, lines PQ and RS are parallel and the lines l and m are parallel.
Perpendicular Lines
Perpendicular lines are lines that intersect at right angles. If two lines are perpendicular to each other, then the product of their slopes is equal to - 1. In the figure shown below, the lines AB and EF are perpendicular to each other.
Like terms
Probably the most common thing you will be doing with polynomials is "combining like terms". This is the process of adding together whatever terms you can, but not overdoing it by trying to add together terms that can't actually be combined. Terms can be combined ONLY IF they have the exact same variable part. Here is a rundown of what's what: 4x and 3 NOT like terms The second term has no variable 4x and 3y NOT like terms The second term now has a variable, but it doesn't match the variable of the first term 4x and 3x2 NOT like terms The second term now has the same variable, but the degree is different 4x and 3x LIKE TERMS Now the variables match and the degrees match Once you have determined that two terms are indeed "like" terms and can indeed therefore be combined, you can then deal with them in a manner similar to what you did in grammar school. When you were first learning to add, you would do "five apples and six apples is eleven apples". You have since learned that, as they say, "you can't add apples and oranges". That is, "five apples and six oranges" is just a big pile of fruit; it isn't something like "eleven applanges". Combining like terms works much the same way. Simplify 3x + 4x These are like terms since they have the same variable part, so I can combine the terms: three x's and four x's makes seven x's: Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved 3x + 4x = 7x Like Terms are terms whose variables (and their exponents such as the 2 in x2) are the same. In other words, terms that are "like" each other. (Note: the coefficients can be different) Example: (1/3)xy2 -2xy2 6xy2 Are all like terms because the variables are all xy2
Reflex Angles
Reflex Angle is an angle that lies between 180° and 360°. More about Reflex Angle A full angle is also a reflex angle. Which among the following represents a reflex angle? Choices: A. Figure 1 B. Figure 2 C. Figure 3 D. Figure 4 Correct Answer: C Solution: Step 1: Reflex Angle is an angle that lies between 180° and 360°. Step 2: Here, the angle in Figure 3 is more than 180° and less than 360°. Step 3: So, the angle in Figure 3 is a reflex angle.
Space
Space exists in three dimensions. Space is made up of all possible planes, lines, and points. It extends indefinitely in all directions.
Square
Square is a parallelogram with right angles and equal sides ( Fig.35 ). A square is a particular case of a rectangle and a rhombus simultaneously; so, it has all their above mentioned properties.
Subtraction Property of Equality
Subtractive Property of Equations: If the same number is subtracted from both sides of an equation, then the two sides remain equal. That is, if x = y, then x - z = y - z.
Surface Area
Surface Area Formulas In general, the surface area is the sum of all the areas of all the shapes that cover the surface of the object. Surface Area of a Cube = 6 a 2 Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac (a, b, and c are the lengths of the 3 sides) Surface Area of a Sphere = 4 pi r 2 (r is radius of circle) Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h (h is the height of the cylinder, r is the radius of the top) Surface Area = Areas of top and bottom +Area of the side Surface Area = 2(Area of top) + (perimeter of top)* height Surface Area = 2(pi r 2) + (2 pi r)* h
Circumference
The Circumference is the distance around the edge of the circle. Circumference = π × Diameter The length of the words may help you remember: Radius is the shortest word Diameter is longer (and is 2 × Radius) Circumference is the longest (and is π × Diameter)
Diameter
The Diameter starts at one side of the circle, goes through the center and ends on the other side.
Exterior Angles
The Exterior Angle is the angle between any side of a shape, and a line extended from the next side. Note: When you add up the Interior Angle and Exterior Angle you get a straight line, 180°. (See
Inverse Property of Multiplication
The Inverse Property of Multiplication states that the product of any number and its multiplicative inverse is 1. Inverse property of multiplication states that every non-zero number "a", when multiplied by "1/a" gives 1 as the answer. Here 1 is the identity element for multiplication and 1/a of the number is inverse of multiplication. Formula : a * (1/a) = 1 Example : 5 * 1/5 = 1
Radius
The Radius is the distance from the center to the edge.
Area
The area of a circle is π times the radius squared, which is written: A = π × r2 Or, in terms of the Diameter: A = (π/4) × D2 Example: What is the area of a circle with radius of 1.2 m ? A = π × r2 A = π × 1.22 A = π × (1.2 × 1.2) A = 3.14159... × 1.44 = 4.52 (to 2 decimals)
Perimeter
The perimeter of any polygon is the sum of the lengths of all the sides. square = 4a rectangle = 2a + 2b triangle = a + b + c circle = 2pi r circle = pi d (where d is the diameter) The perimeter of a circle is more commonly known as the circumference.
Symmetry
The simplest symmetry is Reflection Symmetry (sometimes called Line Symmetry or Mirror Symmetry). It is easy to recognise, because one half is the reflection of the other half.
Inverse Property of Addition
The sum of a number and its additive inverse is always zero. (x + (-x) = 0)
Exponent
This process of using exponents is called "raising to a power", where the exponent is the "power". The expression "53" is pronounced as "five, raised to the third power" or "five to the third". There are two specially-named powers: "to the second power" is generally pronounced as "squared", and "to the third power" is generally pronounced as "cubed". So "53" is commonly pronounced as "five cubed". When we deal with numbers, we usually just simplify; we'd rather deal with "27" than with "33". But with variables, we need the exponents, because we'd rather deal with "x6" than with "xxxxxx". Exponents have a few rules that we can use for simplifying expressions. Simplify (x3)(x4) Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved To simplify this, I can think in terms of what those exponents mean. "To the third" means "multiplying three copies" and "to the fourth" means "multiplying four copies". Using this fact, I can "expand" the two factors, and then work backwards to the simplified form: (x3)(x4) = (xxx)(xxxx) = xxxxxxx = x7 Note that x7 also equals x(3+4). This demonstrates the first basic exponent rule: Whenever you multiply two terms with the same base, you can add the exponents: ( x m ) ( x n ) = x( m + n ) However, we can NOT simplify (x4)(y3), because the bases are different: (x4)(y3) = xxxxyyy = (x4)(y3). Nothing combines.
Equilateral Triangle
Three equal sides Three equal angles, always 60°
Complimentary Angles
Two Angles are Complementary if they add up to 90 degrees (a Right Angle). These two angles (40° and 50°) are Complementary Angles, because they add up to 90°. Notice that together they make a right angle. But the angles don't have to be together. These two are complementary because 27° + 63° = 90°
Supplementary Angles
Two Angles are Supplementary if they add up to 180 degrees. These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°. Notice that together they make a straight angle. But the angles don't have to be together. These two are supplementary because 60° + 120° = 180°
Adjacent Angles
Two angles are Adjacent if they have a common side and a common vertex (corner point) and don't overlap. Angle ABC is adjacent to angle CBD Because: they have a common side (line CB) they have a common vertex (point B)
Isosceles Triangle
Two equal sides Two equal angles
Opposites
Two numbers that have the same magnitude but are opposite in signs are called Opposite Numbers. More about Opposite Numbers When opposite numbers are added, it gives zero. To get the opposite of a number, change the sign. The absolute values of opposite numbers are the same. The opposite numbers are equidistant from 0 on a number line. Examples of Opposite Numbers + 25 and - 25 are the opposite numbers. - 8 is the opposite number of + 8.
Terms
Variables
Vertical Angles
Vertical Angles are the angles opposite each other when two lines cross "Vertical" in this case means they share the same Vertex (corner point), not the usual meaning of up-down. In this example, a° and b° are vertical angles. The interesting thing here is that vertical angles are equal: a° = b° (in fact they are congruent angles)
Alternate Interior Angles
When two lines are crossed by another line (called the Transversal): The pairs of angles on opposite sides of the transversal but inside the two lines are called Alternate Interior Angles. In this example, these are Alternate Interior Angles: c and f are Alternate Interior Angles d and e are also Alternate Interior Angles (To help you remember: the angle pairs are on "Alternate" sides of the Transversal, and they are on the "Interior" of the two crossed lines)
Opposite Interior Angles
When two lines are crossed by another line (called the Transversal): The pairs of angles on opposite sides of the transversal but inside the two lines are called Alternate Interior Angles. In this example, these are Alternate Interior Angles: c and f are Alternate Interior Angles d and e are also Alternate Interior Angles (To help you remember: the angle pairs are on "Alternate" sides of the Transversal, and they are on the "Interior" of the two crossed lines)
Parallelogram
is a quadrangle, opposite sides of which are two-by-two parallel. Any two opposite sides of a parallelogram are called bases, a distance between them is called a height ( BE, Fig.32 ). Properties of a parallelogram. 1. Opposite sides of a parallelogram are equal ( AB = CD, AD = BC ). 2. Opposite angles of a parallelogram are equal ( A = C, B = D ). 3. Diagonals of a parallelogram are divided in their intersection point into two ( AO = OC, BO = OD ). 4. A sum of squares of diagonals is equal to a sum of squares of four sides: AC² + BD² = AB² + BC² + CD² + AD² . Signs of a parallelogram. A quadrangle is a parallelogram, if one of the following conditions takes place: 1. Opposite sides are equal two-by-two ( AB = CD, AD = BC ). 2. Opposite angles are equal two-by-two ( A = C, B = D ). 3. Two opposite sides are equal and parallel ( AB = CD, AB || CD ). 4. Diagonals are divided in their intersection point into two ( AO = OC, BO =
Rotational Symmetry (2 dimensions)
With Rotational Symmetry, the image is rotated (around a central point) so that it appears 2 or more times. How many times it appears is called the Order. Here are some examples (they were made using Symmetry Artist, and you can try it yourself!) When we rotate the triangle about its center point for 360o, we will notice that it fits onto itself for 3 times for every 120o rotation. By definition, the number of times a shape fits onto itself when rotated is called the order of symmetry. Hence, we can see that the order of symmetry for this triangle is 3.
Order of Rotational Symmetry
With Rotational Symmetry, the image is rotated (around a central point) so that it appears 2 or more times. How many times it appears is called the Order. When we rotate the triangle about its center point for 360o, we will notice that it fits onto itself for 3 times for every 120o rotation. By definition, the number of times a shape fits onto itself when rotated is called the order of symmetry. Hence, we can see that the order of symmetry for this triangle is 3.
Plane of Symmetry
You can find if a shape has a Line of Symmetry by folding it. When the folded part sits perfectly on top (all edges matching), then the fold line is a Line of Symmetry. Above is NOT A LINE OF SYMMETRY Here I have folded a rectangle one way, and it didn't work.
