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In addition to geometry theorems that have converses, there are statements and assertions which have converses. Two examples are explained here in the form of pairs of statements.
Example 1. (a) If a number is even, (b) then it is exactly divisible by 2, without a remainder.
The converse assertion reverses these statements: (a) If a number is exactly divisible by 2 without a remainder, (b) then the number is even.
Both the statement and its converse are true.
Example 2. The converse of the following is not necessarily true: (a) If A is the father of B, (b) then B is the child of A.
The converse states that (a) If B is the child of A, (b) then A is the father of B.
CONVERSION FACTOR 115
The converse of this statement is not necessarily true, because A could be the mother of B and not the father.
References: Circle Geometry Theorems, Cyclic Quadrilateral, Geometry Theorems, Pythagoras’ Theorem.
Conversions are used to change the units of a quantity. For example, the units for measuring temperature are degrees Celsius (°C) and degrees Fahrenheit (°F). It may be necessary to change the units from one to the other. In order to change the units, a formula is needed, as illustrated in the following example.
Example. The temperature of a warm, sunny day is 24°C. What temperature is this on the Fahrenheit scale?
F = | C + 32 This is the formula for converting from °C to °F
F = | x 24 + 32 Substituting C = 24
F = 75.2°F Using the calculator
References: Currency Conversions, Temperature.
The idea of a conversion factor is explained using the following example.
Madge owns a ladies dress shop and is having a winter sale. She decides to reduce all items of clothing by 10%. This may be referred to as giving a discount of 10%. A dress that usually sells for $124 will be reduced in price by 10%. Write
Discount = 10% of $124
10 „ 10
= —- x $124 10% = —-
= $12.40 Using the calculator
The discount is the price reduction of the dress, and this reduction is $12.40. The sales price is $124 — $12.40 = $111.60. This sales price can be obtained in one step by multiplying the normal price of $124 by the figure 0.9:
$124 x 0.9 =$111.60
This multiplying figure of 0.9 is called the conversion factor for converting the normal price of goods to a sale price.
References: Conversion, Discount, Percentage.
Reference: Cartesian Coordinates.
Points, lines, and polygons are coplanar if they lie in the same plane. Lying in the same plane means lying in the same flat surface. The problem about whether or not points, lines, and polygons are coplanar only arises in three-dimensional figures, where more than one plane exists. The figure of a three-dimensional house is used to explain the term coplanar.
Coplanar points: The points A, B, C, and D in the figure are coplanar, because they all lie in the same plane, which is the sloping roof. The points A, B, C, and F are not coplanar, because the point F does not lie in the same plane as the other points A, B, and C.
Coplanar lines: The line segments AB and CD are coplanar, because they lie in the same plane, which is the sloping roof of the house. DC and EF are also coplanar, and their common plane is the wall. In fact the line segments AB and EF are also coplanar, and their common plane is ABFE, which is an inclined plane and is not drawn in the figure. The three line segments AB, CD, and EF are not all coplanar.
Coplanar polygons: The two square windows are coplanar polygons, and their common plane is the wall of the house, but they are not coplanar with the rectangular door.
Skew lines: When lines that are not parallel and not coplanar they are called skew lines. The lines AD and CF are skew lines. Skew lines do not intersect and are not parallel.
References: Line, Line Segment, Parallel, Plane, Point, Polygon.
This is an additional theorem that follows from a main theorem that has already been proved. As an example, consider the angle at the center theorem, which is a circle geometry theorem. The main theorem is (see figure a): The angle subtended at the center O of a circle by the arc AB is equal to twice the angle that is subtended by the same arc at any point C on the circumference of the same circle. In symbols what this means is that
Angle AOB = 2 x angle ACB
The corollary of this main theorem is as follows: Suppose the minor arc AB is made longer so that it becomes a semicircle (see figure b). Then the angle AOB at the center is now 180°, and the angle ACB at the circumference will be 90°. The corollary of the angle at the center theorem is now: The angle in a semicircle is a right angle.
Reference: Geometry Theorems.
There are a number of terms, listed below, which are related and can best be explained together under this entry: