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9. Geometry
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Middle Years

9.01 Geometrical diagrams

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Geometry is the study of shapes, and is one of the oldest areas of mathematical interest. Geometrical diagrams have many important features, and the terminology used in the subject is important to ensure good mathematical communication.

Points, segments, rays, lines

A point is a single location, with no height or width. We use capital letters to distinguish two different points. This diagram shows three different points labelled $A$A, $B$B, and $C$C:

If we connect all of the points from one point to another, we make a segment. This is the segment between the points $A$A and $B$B:

A segment always contains its endpoints. We will sometimes say it is the segment from $A$A to $B$B, which is the same as the segment from $B$B to $A$A. We will often use the abbreviation $AB$AB to mean this segment.

If we start at one point and keep going, we make a ray. This is the ray from $A$A through $B$B:

And this is the ray from $B$B through $A$A:

Direction is important for rays - these two objects are not the same!

If we keep going in both directions, we make the line through $A$A and $B$B:

To summarize, segments and lines stay the same if we reverse the order of the points, but this is not true for rays:

Explore these applets to investigate these objects further. Make a selection and drag the coloured point to highlight all the points that lie on that object:

Segments Lines
Rays

We place small markings on segments when we want to show that they are equal in length. In this diagram, the segment $AB$AB has the same length as the segment $AC$AC:

This does not mean that the two segments are made up of the same points - only that they have the same length. Sometimes we will use more than one kind of marking to show that some segments are equal to others. In this diagram we use both one-stroke markings and two-stroke markings:

Move the points around in this applet to see how the segments stay the same length:

Summary

A line between two points contains every point between the points and all the points beyond on either side. A ray starts at one point and continues through another and beyond. A segment starts at one point and stops at the other.

 

Angles

Whenever two lines, rays, or segments pass through the same point, we can describe the relative orientation of one to the other using an angle. Here are two rays drawn from the same point through two other points:

There are two ways to turn from one to the other. The shorter turn is simply called the angle between the objects, and the larger turn is called the reflex angle. We draw a circular arc from one object to the other to denote the angle (or reflex angle):

We can use three points to refer to an angle by using the symbol "$\angle$∠" followed by three letters, one for each point. The first letter will be on one of the rays, lines, or segments, the second point will be their intersection, and the third will be a point on the other ray, line, or segment. This means there are two equally valid ways to refer to an angle, as illustrated in this diagram:

Just like with segments, we can use additional markings to show that two angles are equal. We draw multiple arcs to show that different angles are equal to each other. In this diagram the two angles drawn with double arcs are equal:

Summary

The angle between two intersecting segments, lines, or rays represents their relative orientation to each other. We write the angle symbol "$\angle$∠" followed by three letters. The second letter is always the intersection point, and the first and third letters lie on the objects forming the angle, one on each.

Swapping the first and third letters does not change the angle.

 

Practice questions

Question 1

Select the diagram that shows the line through $E$E and $F$F:

  1. The image shows a straight line with arrows at both ends, indicating that it extends infinitely in both directions passing through the points F and G on the line, characteristic of a geometric line. There are three labeled points on this image: point E, point F, and point G. Point F and point G is marked with a dot on the line, and point E is marked with a dot but is not directly on the line, suggesting it is not collinear with points F and G.

    A

    The image shows a straight line with dots at both ends, indicating that the line starts from point E and stops at point F on the line, characteristic of a geometric segment. There are three labeled points on this image: point E, point F, and point G. Point E and point F is marked with a dot on the segment, and point G is marked with a dot but is not directly on the segment, suggesting it is not collinear with points E and F.

    B

    The image shows a straight line with arrows at both ends, indicating that it extends infinitely in both directions passing through the points E and F on the line, characteristic of a geometric line. There are three labeled points on this image: point E, point F, and point G. Point E and point F is marked with a dot on the line, and point G is marked with a dot but is not directly on the line, suggesting it is not collinear with points E and F.

    C

    The image shows a straight line with dots at both ends, indicating that the line starts from point E and stops at point G on the line, characteristic of a geometric segment. There are three labeled points on this image: point E, point F, and point G. Point E and point G is marked with a dot on the segment, and point F is marked with a dot but is not directly on the segment, suggesting it is not collinear with points E and G.

    D

    The image shows a straight line with an arrow at one end, indicating that it extends infinitely in one direction starting from point F passing through point E on the line, characteristic of a geometric ray. There are three labeled points on this image: point E, point F, and point G. Point E and point F is marked with a dot on the ray, and point G is marked with a dot but is not directly on the ray, suggesting it is not collinear with points E and F.

    E

    The image shows a straight line with an arrow at one end, indicating that it extends infinitely in one direction starting from point F passing through point G on the line, characteristic of a geometric ray. There are three labeled points on this image: point E, point F, and point G. Point F and point G is marked with a dot on the ray, and point E is marked with a dot but is not directly on the ray, suggesting it is not collinear with points F and G.

    F
Question 2

This diagram has the angle $\angle ABC$∠ABC marked. What is another way of referring to the same angle?

An angle with vertex B, which arc is shaded in violet, is formed by two line segments each starting from two different points and converging to a one central point. One is line segment AB, formed by connecting Point A and the central point B. The other one is line segment CB, formed by Point C and the central point B.

  1. $\angle BAC$∠BAC

    A

    $\angle ACB$∠ACB

    B

    $\angle CBA$∠CBA

    C
Question 3

This diagram has two equal segments marked:

  1. What segment is equal in length to $ZY$ZY?

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