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The A to Z of mathematics a basic guide - Sidebotham T.H.

Sidebotham T.H. The A to Z of mathematics a basic guide - Wiley publishing , 2002. - 489 p.
ISBN 0-471-15045-2 Previous << 1 .. 2 3 4 < 5 > 6 7 8 9 10 11 .. 126 >> Next 8 ACUTEANGLE
ACUTE ANGLE
This entry also discusses right angle, straight angle, obtuse angle, and reflex angle. In order to define acute angles it is first necessary to explain a right angle. An angle of 90 degrees, which is written as 90Â°, is called a right angle. A right angle is indicated in the following diagrams by a box, and represents a quarter turn. A flagpole makes a right angle with the ground (see figure a).
Angles that are less than one right angle, that is, less than 90Â°, are called acute angles. If the lid of a box is opened through an acute angle and then let go, the lid will fall back onto the top of the box (see figure b).
Every triangle must have at least two acute angles (see figure c).
An angle of 180Â° is called a straight angle and corresponds to a half turn.
On a clock face when the time is 6 oâ€™clock, the angle between the two hands is a straight angle (see figure d).
An obtuse angle is greater than 90Â°, but less than 180Â°. If the lid on a box is opened through an obtuse angle and let go, it will fall open and not fall back onto the top of the box (see figure e).
Right
angle
(b)
(e)
(f)
The trapezium shown in figure f has two obtuse angles.
A reflex angle is greater than 180Â°, but less than 360Â°. An example of a reflex angle is a three-quarters turn. When the lid of a box is opened fully, the lid has turned through a reflex angle (see figure g).
When a quadrilateral contains a reflex angle, it is called a reentrant quadrilateral (see figure h).
More than two addends might be involved. For example, to get the sum of 3 + 9 + 6; we add together the pair of numbers 3 and 9 to get 12, and then add the 12 to 6 to get 18. Or, in adding numbers in our head, we might develop the skill of first arranging them into pairs that add to make 10, because it is easy to add numbers onto 10. For example,
(9)
(h)
7 + 8= 7 +(3 + 5) Splitting 8 up into 3 + 5
= (7 + 3) + 5 Combining the 7 with the 3 to make 10
= 15
Reference: Associative Law.
Angles such as a: and y in figure a that lie side by side are called adjacent angles.
10 ALGEBRA
Any number of adjacent angles that form a straight line add up to 180Â°. When two adjacent angles form a straight line they add up to 180Â°. In figure b, angles a and b are supplementary angles:
a + b = 180Â°
Example 1. If a spade makes an acute angle of 76Â° with the horizontal ground, find the obtuse angle that the spade makes with the ground.
Solution. Let this obtuse angle be x (see figure c):
x + 76Â° = 180Â° Sum of adjacent angles = 180Â°
x = 104Â° Subtracting 76Â° from both sides of the equation to solve it
The spade makes an obtuse angle of 104Â° with the ground.
Example 2. Light rays are reflected from a mirror as shown in figure d. Find the angle x between the two rays.
Solution. The calculation goes as follows:
35Â° + x + 35Â° = 180Â° Sum of adjacent angles = 180Â°
jc + 70Â° = 180Â° Simplifying
x = 110Â° Subtracting 70Â° from both sides of the equation to
solve it
Reference: Geometry Theorems.
ALGEBRA
Algebra is the abstract study of the properties of numbers, using letters to stand for the numbers; these letters are called variables. Variables stand for unknown quantities and
ALGEBRA 11
we use the operations of arithmetic to try to find their value. This entry explains how to simplify, or rewrite, expressions by adding, subtracting, multiplying, and dividing terms.
Example 1. Bill is doing a preliminary sketch of the ground floor of a house he is designing. He is not sure of some of the dimensions and uses variables x and y to represent them. All measurements are in meters. His sketch is shown in figure a, and is not to scale. Find the perimeter of the house.
Solution. The perimeter is the distance all the way around the house:
Perimeter = x + 3y + 8 + y+18 + 3x
Perimeter = x + 3x + 3y+y + 8+18 Grouping similar terms together
Perimeter = 4x + 4y + 26 meters Adding similar terms together.
This expression for the perimeter cannot be simplified further.
3x
3y
18
(a)
When terms are similar and can be simplified by adding or subtracting, we say they are â€ślike terms.â€ť The process of adding and subtracting like terms is called â€ścollecting terms.â€ť Terms are like terms if they are exactly the same except for the number in front of them. This number written in front of the term is called the coefficient of the term. For example, the coefficient of y in the term â€”3y is â€”3.
Examples of sets of like terms are (i) 2a, 4a, â€”6a, 24a; (ii) xy, 3xy, 5xy, â€”14xy; (iii) 4x2,x2, 2x2, â€”16.x2.
Example 2. Simplify these expressions: (i) â€”2xy + 4xy â€” 3xy, (ii) a + 2b â€” 2 a + ab.
Solution. For part (i)
â€”2xy + 4xy â€” 3xy = â€” 1 xy The terms are all like terms, and calculating â€” 2 +
4 â€” 3 = â€” 1 gives the coefficient of xy Previous << 1 .. 2 3 4 < 5 > 6 7 8 9 10 11 .. 126 >> Next 