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9.01 Geometrical diagrams

Lesson

Introduction

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 and segments

A point is a single location, with no height or width. We use capital letters to distinguish two different points.

3 points named  A, B, and C.

This diagram shows three different points labelled A, B, and C.

If we connect all of the points from one point to another, we make a segment.

Points A, B, and C where A and B are connected forming a line segment.

This is the segment between the points A and B.

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

Exploration

Explore the applet to investigate segments. Make a selection and drag the coloured point to highlight all the points that lie on that segment:

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A segment is a composition of points connected together between two endpoints.

We place small markings on segments when we want to show that they are equal in length.

Segments AC and AB are connected through point A. Each segment has one-stroke markings so they have equal length.

In this diagram, the segment AB has the same length as the segment 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.

Segments AB and BC have two-stroke markings while segments CD and DE have one-stroke markings. AB equals BC and CD equals DE.

In this diagram we use both one-stroke markings and two-stroke markings.

Examples

Example 1

This diagram has two equal segments marked:

Segments XW, ZX, ZY and XY. Segments XW and ZY have one-stroke markings.

What segment is equal in length to ZY?

Worked Solution
Create a strategy

Choose the segment with the same marking as the given segment.

Apply the idea

Segment XW has the same marking as ZY.

So segment XW is equal in length to ZY.

Idea summary

A point is a single location, with no height or width and can be labelled as a single letter like A or B.

A segment is a composition of points connected together between two endpoints.

Segments AB and BC have two-stroke markings while segments CD and DE have one-stroke markings. AB equals BC and CD equals DE.

We place small markings to show the segments that are equal in length.

Rays

If we start at one point and keep going, we make a ray.

Points A, B, and C where A and B are connected with arrow at B forming a downward ray.

This is the ray from A through B.

Points A, B, and C where A and B are connected with arrow at A forming a upward ray.

And this is the ray from B through A.

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

Exploration

Explore the applet to investigate rays. Make a selection and drag the coloured point to highlight all the points that lie on that ray:

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The ray extends through the second point in the name of ray. For instance, for Ray {AB}, the ray extends through B, not A.

Idea summary

A ray is a composition of points connecting two points that goes through one of those points.

The ray extends through the second point in the name of ray. For instance, for Ray {AB}, the ray extends through B, not A.

Lines

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

Points A, B, and C where A and B are on a line.

Exploration

Explore the applet to investigate lines. Make a selection and drag the coloured point to highlight all the points that lie on that line:

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A line is a composition of points that passes through two points.

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

This image shows the difference of rays from segments and lines. Ask your teacher for more information.

Examples

Example 2

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

A
Point E and line FG
B
Segment EF and point G
C
Line EF and point G
D
Segment EG and point F
E
Ray FE and point G
F
Point E and ray FG
Worked Solution
Create a strategy

The line should pass through both points and have arrows at both ends.

Apply the idea

We need to choose an option where an extending line passes through points E and F.

The answer is option C.

Idea 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.

Image containing a line, ray, and segment.

Rays go through different directions when the points are reversed which makes it different from segments and lines.

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.

Image of two rays from the same point to form an angle.

Here are two rays drawn from the same point through two other points.

Image of two rays from the same point to form an angle on the right and a reflex angle on the left.

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:

Two equal angles. The first is named angle B A C and second is named angle C A B. Ask your teacher for more information.

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.

Angles B A C and D C E have double arcs and so are equal. Angle A C F has triple arcs and angle F C E has a single arc.

In this diagram the two angles drawn with double arcs are equal.

Examples

Example 3

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

Angle A B C where point B is the intersection of segments A B and B C.
A
\angle {BAC}
B
\angle {ACB}
C
\angle {CBA}
Worked Solution
Create a strategy

We can reverse the letters in the original name to find a new name for the angle.

Apply the idea

Reversing the points in \angle {ABC} gives us \angle {CBA}, so option C.

Idea 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.

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