Adjacent angles can be added or subtracted.The following reasons can be used in geometric arguments: If two lines intersect so that all four angles are right-angles, then the lines are said to be perpendicular. ![]() The argument above is a proof of the theorem sometimes proofs are presented formally after the statement of the theorem. A theorem is an important statement which can be proven by logical deduction. We thus have our firstĪ result in geometry (and in mathematics generally) is often called a theorem. We can conclude that these vertically opposite angles,ĪOX and BOY are equal. BOY is also the supplement of BOX (straight angle),.AOX is the supplement of BOX (straight angle).In the diagram, the angles marked AOX and BOY are called vertically opposite. When two lines intersect, four angles are formed at the point of intersection. Hence, in the diagram, AOB and BOC are adjacent.Īdjacent angles can be added, so in the diagram Two angles at a point are said to be adjacent if they share a common ray. Three (or more) lines that meet at a single point are called concurrent. Three (or more) points that lie on a straight line are called collinear. In the second diagram, we write AB || CD. Lines which never meet are called parallel. Given two distinct lines, there are two possibilities: They may either meet at a single point or they may never meet, no matter how far they are extended (or produced). Thus, the given line below is referred to as the line AB or as the line. We use capital letters to refer to points and name lines either by stating two points on the line, or by using small letters such as and m. Given two distinct points A and B then there is one (and only one) line which passes through both points. When we draw a line it has width and it has ends, so it is not really a line, but represents a line in our imagination. A line has no width and extends infinitely in both directions. In practice, when we draw a point it clearly has a definite width, but it represents a point in our imagination. A point marks a position but has no size. Because they are so simple, it is hard to give precise definitions of them, so instead we aim to give students a rough description of their properties which are in line with our intuition. The simplest objects in plane geometry are points and lines. Thus geometry gives an opportunity for students to develop their geometric intuition, which has applications in many areas of life, and also to learn how to construct logical arguments and make deductions in a setting which is, for the most part, independent In secondary school, the level of rigour should develop slowly from one year to the next, but at every stage clear setting out is very important and should be stressed. In secondary school geometry, we begin with a number of intuitive ideas (points, lines and angles) which are not at all easy to precisely define, followed by some definitions (vertically opposite angles, parallel lines, and so on) and from these we deduce important facts, which are often referred to as theorems. Just as arithmetic has numbers as its basic objects of study, so points, lines and circles are the basic building blocks of plane geometry. Geometry also has many applications in art. Classifying such geometric objects and studying their properties are very important. A view of the roofs of houses reveals triangles, trapezia and rectangles, while tiling patterns in pavements and bathrooms use hexagons, pentagons, triangles and squares.īuilders, tilers, architects, graphic designers and web designers routinely use geometric ideas in their work. Geometry is used to model the world around us.
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