You plan to change your room a little bit and develop your own style. You have already bought a seating area from your trusted furniture store that is at an angle of 110° to each other. In order to place a cupboard next to it on the wall, in which a music system belongs, you need about 80°. Can you put both pieces of furniture right next to each other?
For this task, knowledge about side winkotherwise necessary.
Minor angles in mathematics belong to the various angles between two intersecting lines. In this explanation you will find out how you can calculate these and what the context is for other angles.
Secondary Angle – Repeat Angle
In principle, angles can occur if intersect two straight lines respectively if three points make an angle.
if two straights itself cutthen there are four in total in the geometry angle. The size of an angle is specified in degrees (°).
It is important that when specifying the angle, always include the direction counterclockwise is chosen.
angles can on two different types develop:
- between two intersecting lines ()
- between three points, where the vertex – the point at the angle – is written in the middle ()
Figure 1: Angle with specification for two straight lines and three points
Special angular sizes are mainly at 0° or 90°. All other angle sizes are classified as follows:
A vertex angle occurs at two intersecting lines. Of the four resulting angles, the opposite angles are called the vertex angles.
In Figure 3 there are four different angles, namely . They were arranged counterclockwise, with vertex angle are.
Figure 3: Apex angle for two intersecting straight lines
If two straight lines run parallel and another straight line intersects these two, other types of angles can arise, such as the alternating angle or that step angle. Please have a look at these explanations if you would like to learn more about them.
Secondary Angle – Properties
At the beginning it will be explained to you what secondary angles are and how you can deduce them. There is a small derivation that should help you to understand side angles well.
Secondary angle – explanation and derivation
Secondary angles arise when two straight lines g and h meet point P cut.
At the point of intersection P there is then one line crossing. This divides the surrounding area into four parts, which are delimited by the two straight lines g and h. For example, it looks like this:
Figure 4: Line crossing of two straight lines
As you already know, for a straight line, is one straight angle 180°. This straight angle is referred to below as .
ε is the Greek letter for e and is pronounced «epsilon».
derivation
Figure 5: Line crossing with straight angle
In order to better understand the principle of the secondary angle, the angle along the straight line g was entered in the diagram.
From this picture you can see that the sum of angles from the angle shown in figure 8 and together in figure 7 gives the angle (180°).
So the following applies:
Graphically it looks like this:
Figure 6: Secondary angle at a line crossing with a total of 180°
The angles and are therefore secondary angles. They are next to each other at the intersection.
In general, it helps to visualize the two crossing lines.
secondary angle occur when two straight lines intersect. Two angles are side angles of each other when they are at the intersection of the straight lines side by side lie.
Associated secondary angles are also called pair of angles designated. Altogether arise at the intersection of two straight lines four pairs of angles.
Figure 9: Four pairs of secondary angles
The so-called law of secondary angles applies to these pairs of secondary angles.
Secondary Angle Theorem and Secondary Angle Pairs
It has already been implied that overall four different pairs of angles are. They always join one line crossing on, whereby the following conditions apply:
secondary angle theorem
The angle sum of secondary angles at a line crossing is always 180°. Side angles therefore always complement each other to form one straight angle.
The opposite pairs of angles are:
The other two pairs of angles are:
These are shown again in the graphic below:
Figure 10: Four pairs of angles at the crossing of straight lines
The two angles are each commutative, which means you can also enter them interchangeably. Both statements are therefore identical:
What exactly does «commutative» mean? You can read about that again in the article on the commutative law.
Calculate secondary angle
So you have already been able to deduce how the secondary angles work and that they add up to 180°.
Use the example above to calculate the secondary angle specifically. So you know that two secondary angles together have 180°. By specifying one of these angles, you can deduce the other.
For example, there are two angles. These are the exact values.
Both taken together result in 180°.
Now assume that only is given. You can use this to close the gap by subtracting the one secondary angle from the 180°.
This means that when you specify an angle, you can use the automatically second specification of 180° to infer the third angle.
Task 1
From the introduction you learned that you bought a corner sofa that is 110° in total. You want to place a cupboard next to it, for which you need 80°. Can you place both pieces of furniture right next to each other?
solution
So you gave an angle, namely 110°. The second angle is the wall itself, which is itself flat, making a total of 180°. You can now secondary angle determine.
For example, denote the given angle as and the desired one as .
This means that you have a total angle of 70° available for the piece of furniture. However, you need 80°, that’s 10° too much. That’s why you can’t put the pieces of furniture right next to each other.
But it may also be the case that you should first determine one angle yourself with the protractor in order to calculate the other. If you would like to know how to determine angles with a protractor again, you can go to the explanation measure angle or Calculate angle stop by.
side angle triangle
Secondary angles can also play an important role in the triangle when it comes to the so-called interior angles and the sum of the interior angles in the triangle.
Look at the graphic below, you will see how minor angles can be determined with a straight line drawn through the vertices of a triangle:
Figure 11: Minor angles in purple for interior angles of the triangle (in pink)
So that means you can use the secondary angle to determine the interior angle of a triangle.
The angle is given as 135°. With this you can use the secondary angle theorem determine the angle:
You can now use this result for the so-called interior angle sum in the triangle.
If you have now calculated a second interior angle of the triangle, you can also use the sum of the interior angles in the triangle to determine the third angle.
In a triangle, all angles add up to 180°.
You now know two pieces of information:
Now you can close by entering the values for the angles and solving for the angle you are looking for.
You can find more information in the declaration Sum of interior angles triangle or the opposite of that: Sum of the exterior angles of a triangle.
Secondary Angle – Examples & Tasks
Finally, in this section you can practice everything you have learned about secondary angles in this article with some examples.
Secondary angle – the most important things at a glance
- if two straights intersect, arise as a whole four anglewhere opposite angles are called vertex angles.
- secondary angle occur when two straight lines intersect. If two angles are secondary angles of each other, they are also called side angle pair designated.
- At an intersection of two straight lines arise altogether four pairs of angles.
- the sum of angles of a pair of secondary angles is always 180°.
- If α and β are secondary angles, the following applies: .
proof
- Erbrecht et al. (2012). The large table work interactive Formula collection for the secondary level I and II. Cornelsen Verlag, Berlin