Geometry is an area that many in Sweden often want to stay as far away from as possible. This image of being a terrible subject is in desperate need of change. In fact, it is a very nice subject and the biggest delight with it is that with quite little you can get very far and even solve problems at a very high level. In this first section on angles, we will provide the first cornerstone needed to succeed in this.

Basically, one can summarize the area as a set of angles and distances that can be combined together to form different geometric shapes. A triangle is an example of a geometric figure consisting of $3$ angles and $3$ distances (= sides) and correspondingly a square is a geometric figure consisting of $4$ angles and $4$ sides. However, the angles and sides cannot be chosen freely, as you may recognize, the sum of the angles in a triangle must be $180 ^{\circ}$ and in the square each angle must be $90^ {\circ}$ and all sides be the same length.

If we start with what an angle really is: an angle is what is formed between two lines and is usually denoted by an angle arc as in the figure below. The unit in which it is measured is degrees and is denoted by a degree sign, $^ {\circ}$. The point from which the angle is based is called a vertex and the two lines that meet at the tip are called rays, or in Swedish "vinkelspets" and "vinkelben" as is denoted in the figure.

A whole turn (ie a circle) corresponds to $360^{\circ}$ and a half turn (ie a straight line) corresponds to $ 180^{\circ} $. Angles are usually denoted by an arc as in the image above, an exception being if they are straight, ie $ 90^{\circ} $. This is the angle that occurs in rectangles and squares and is usually denoted by a square as in the picture below.

In addition to right angles, one usually also distinguishes between acute and obtuse angles (in Swedish "spetsig" and "trubbig" respectively). All three are less than $ 180^{\circ} $, what sets them apart is that acute angles (like the bottom left) are less than $ 90^{\circ} $ and obtuse angles are greater than $ 90^{\circ} $ . In a triangle, a maximum of one angle can be obtuse, can you figure out why?

A protractor can be used to measure angles, as well as to draw them. When it comes to problem solving, however, it is usually not so important that the angle is drawn exactly, but rather that it is a reasonable picture. For example, an angle that you know is $ 30^{\circ} $ large should not be drawn as if it were $ 90^{\circ} $.

In fact, a crossing of two lines gives rise to two different angles; one less than 180 ° (in the picture below denoted $ v $) and one greater than $ 180^{\circ} $ (in the picture below denoted $ 360^{\circ} - v $). Unless otherwise stated, the angle between the two lines in calculations is assumed to be the smaller of the two ($ v $).

A common way to denote angles is by using $ 3 $ points: the starting point, the vertex and the end point. In the picture below you can e.g. type $ \angle ABC = v $. The order $ \angle CBA $ also denotes the same angle. It is important to have the vertex point in the middle, $ \angle BCA $ denotes e.g. not this angle but it denotes the angle between the distances $ BC $ and $ AC $.

To be able to solve problems with angles, we first need certain definitions. The figure below shows four different angles: $ u $, $ v $, $ w $ & $ x $ and three lines: $ \ell_1 $, $ \ell_2 $ & $ \ell_3 $. 

For these:

• $ u $ and $ v $ are called supplementary angles and $ u + v = 180^{\circ} $ (forms a straight line). $ v $ and $ w $ are also supplementary angles in this image. Two right angles are created if the supplementary angles are equal.
• $ u $ and $ w $ are called vertical angles and $ u = w $. Proof of why the angles are equal is left as an exercise to the reader.
• $ u $ and $ x $ are called complementary angles. The angles are formed by one and the same line ($ \ell_3 $) which intersects two other lines $ \ell_1 $ & $ \ell_2 $. The angles are called complementary as they are formed in the "same place" in relation to the points of intersection. If the lines $ \ell_1 $ & $ \ell_2 $ are parallel, i.e. they never intersect no matter how much you extend them, the complementary angles $ u $ and $ x $ are equal ($ u = x $). The reverse is also true: if $ u $ and $ x $ are two equal angles and $ u = x $, then the lines $ \ell_1 $ & $ \ell_2 $ must be parallel.
• Finally, $ w $ and $ x $ are called alternative angles. Proof of why the angles are equal is left as an exercise to the reader.

You probably already know that the sum of the angles of a triangle is $ 180^{\circ} $, but what you probably have not seen before is why this is true. 

In the image above you have a triangle formed by the lines $ \ell_2 $ , $ \ell_3 $ & $ \ell_4 $ with the angles $ a $, $ b $ & $ c $. $ u $, $ v $ & $ w $ are the angles formed when the lines $ \ell_3 $ & $ \ell_4 $ intersect the line $ \ell_1 $ which is parallel to $ \ell_2 $. From the definitions above we can see that $ u $ and $ c $ are complementary angles and so are $ a $ and $ w $. The angles $ v $ and $ b $ are vertical angles. Vertical angles are always equal, so $ b = v $. When the lines intersected ($ \ell_1 $ & $ \ell_2 $) are parallel, complementary angles are equal, so $ a = w $ and $ c = u $. The sum of the angles in the triangle is thus the same as the sum of the angles $ u + v + w $:

$a + b + c = u + v + w$

Since we know that the angles $ u $, $ v $ and $ w $ together corresponds to the angle over a straight line ($ \ell_1 $), i.e. $ 180^{\circ} $, the sum of the angles in a triangle must also be $ 180^{\circ} $!

Now we know why the angle sum of a triangle is $ 180^{\circ} $, but why is the angle sum of a quadrilateral $ 360^{\circ} $? Take the quadrangle $ ABCD $ below as an example.

The easiest proof is to draw a line between two opposite corners. Then two triangles are formed ($ \triangle ABD $ and $ \triangle BCD $). The angles $ b $ and $ d $ are divided into $ u $ & $ b-u $ and $ v $ & $ d-v $, respectively. Each triangle has the angle sum $ 180^{\circ} $, and since all the angles in the two triangles constitute the angle sum of the quadrilateral, the angle sum of the quadrilateral must be: $ 2 \cdot 180^{\circ} = 360^{\circ} $. This is actually enough of a reasoning, but if you want to be very strict you can also structure it:

$a + b + c + d = a + u + (b-u) + c +  v + (d-v) = (a + u + v) + ((b-u) + (d-v) + c) = 180^{\circ} + 180^{\circ} = 360^{\circ}$

Just as you can denote an angle using a symbol, '$ \angle $', and $ 3 $ points, you can similarly denote a triangle on the same way. The sign used then is a small triangle $ \triangle $, which is followed by the three points that make up the corners of the triangle. The triangle below can e.g. be written as $ \triangle ABC $, but also $ \triangle BAC $ or $ \triangle CAB $. Unlike when describing the angle where you must have the vertex in the middle, there is no requirement for the order here.

Distances between two points are usually described with straight lines '$ | $' around the end points of the segment. In the triangle above, e.g. the three sides are written as $ | AB | $, $ | BC | $ and $ | CA | $. Here, too, the order is irrelevant and the sides above can just as easily be written as $ | BA | $, $ | CB | $ and $ | AC | $ respectively.

Two types of triangles you usually want to distinguish from the others are those called isosceles and equilateral triangles. That a triangle is isosceles means that two of the sides of the triangle are equal in length. In return a triangle that is equilateral has all three sides of equal length. The fact that a triangle is equilateral also means that it is isosceles as the requirement that two of the sides are equal in length is met. However, an equilateral triangle does not necessarily have to be equilateral. You can compare it with the fact that a square can also be said to be a rectangle as opposite sides are equal in length and all angles are $ 90^{\circ} $, but that a rectangle does not necessarily have to be a square. 

The fact that a triangle is isosceles or equilateral provides not only information about its sides, but also its angles. In a triangle there is a connection between its angles and sides, as mentioned before these cannot be combined together in just any way. One area you will come across later in math, trigonometry, is commonly used to describe this relationship. For now, it is enough that you accept the following:

If two sides of a triangle are equal, opposite angles are equal.

This can also be shown by means of similarity, which is also an area you will encounter later, albeit a little earlier than trigonometry. In an isosceles triangle, these two equal angles are called base angles. The reverse is also true: i.e. if two angles in a triangle are equal, the triangle must be isosceles and opposite sides must also be equal. For an equilateral triangle, it holds that all three sides are equal and thus all the angles in the triangle with the same reasoning will also be equal. Since the sum of the angles of a triangle is always $ 180^{\circ} $, each angle must therefore be $ \frac {180^{\circ}} {3} = 60^{\circ} $. Just as for an isosceles triangle, the reverse is also true: if a triangle has three angles that are equal (i.e. all angles are $ 60^{\circ} $), the triangle is equilateral and all sides are equal in length.

Some good triangles to know the angles of are those obtained by a half square (i.e. a square with a line drawn through one of the diagonals) and a half equilateral triangle (i.e. an equilateral triangle where one of the heights has been drawn). Both can be described in the figures below as $ \triangle ABD $. Calculating the angles is left as an exercise to the reader. 

When drawing figures, you can do it in different ways, but if you are told to draw a quadrilateral, it is most likely a convex (figure (a)) quadrilateral you draw. The fact that the figure you are drawing is convex means that all angles inside the figure are less than $ 180^{\circ} $. Below you can see a convex quadrilateral and a concave (i.e. at least one angle is greater than $ 180^{\circ} $, see figure (b)). The definition of a convex or concave figure also applies to polygons with more corners than $ 4 $, a triangle is not possible to make concave as the angle sum must always be $ 180^{\circ} $. 

In the majority of the problems you encounter in geometry, you talk about convex polygons. Another common type of polygon you come across is the one called regular. The fact that the polygon is regular means that all sides and all angles are equal. A regular quadrilateral is e.g. same as a square.

One of the simplest methods of solving geometry problems is called angle chasing. It is exactly what it's called and is about looking for angles until you have found the one you are looking for. The problem does not necessarily have to be formulated that you have to determine a certain angle. For example a task where you have to decide that a certain triangle is isosceles can be reformulated to show that the two base angles are equal. Because if the base angles are equal, the sides must be equal in size and thus the triangle must be isosceles.

How, then, should one proceed with such tasks? It is not always that the figure the task describes is given to you from the beginning. If so, the first thing you should do is draw it! As mentioned before, it is not super important that the proportions are accurate. It is enough as long as the figure is reasonably correctly drawn. If you know that an angle you are drawing is acute/right/obtuse, you should draw it as such, but it is not necessary to make it exactly as many degrees as it says in the task. It is not always easy to know what the figure should look like and you may need to try it out as you go. Sometimes even several different images can be true. In most problems below, the figure is given to you from the beginning, although not always complete. However, you will probably need to draw them on a separate piece of paper in order to work with them.

When the basic image is ready, you should read through the information you receive in the assignment and make sure that everything is marked in the image. Two equally large angles or sides can e.g. be marked with a line on them. Just make sure to not mix it up too much, if e.g. there are two pairs of sides that are the same size, one pair can be marked with a line and the other with two so as not to confuse which sides are equal to each other. It can be extra good to write down the information you get next to the figure to double check that you have received everything. It is very easy to draw the figure from the description and then sit and work with it without getting anywhere because you missed an important detail in the description of the task. 

Many angle chasing problems are solved by describing an angle through an equation. Therefore, it may be a good idea to name certain angles in the figure. With that said, you do not have to go overboard and name all the angles you can, it can quickly get messy. However, you can think about what angles it would be relevant to express with a variable and if you need to add more rather do it gradually when you are not getting anywhere. A tip is to start with the one/the ones you are looking for. Some angles can be expressed as a sum or difference of two others (eg $ x = u + v $ from the exterior angle theorem) and you can choose here what you think feels best about giving it a new name ($ x $) and note this ratio, or to simply name it as the sum / difference of the other two angles ($ u + v $). In most cases, it is best to have as few variables as possible to avoid confusion. If you can describe an angle as the sum/difference between two variables, we recommend that you do so instead of introducing another new variable. In cases where you are given the actual angle in degrees, you should of course enter this in the figure directly instead of calling it a variable.

When you have got all the important information from the task and named the relevant angles, you can return to the information you have noted in the figure. If e.g. two sides are equal in size and form two sides in a triangle, you know that this triangle must be isosceles and also the base angles must be equal. An equilateral triangle even gives the angles directly, $ 60^{\circ} $!

In some cases, it is not enough with the image that you get described in the task, but you have to draw extra lines for it to solve itself. This is one of the most difficult things in geometry because it is usually not obvious that you should draw the line(s) until after you have solved the task. In the example below, this is the case. We will show two different ways to solve the task, but of course there are more ways to do it.

Show that if the lines $ \ell_1 $ and $ \ell_2 $ are parallel then the angle $ x = 142^{\circ} $.

The perhaps most important tip for this type of problems is not to panic. It is easy to sit and stare at the picture once it has been drawn and just wait for the answer to pop up and when it does not, you think you are stuck. This is not the case and you usually need more information that you get by working on what you got from the beginning. Instead of waiting for the sudden insight into how to solve the problem, you should instead focus on what you have been given and systematically go through this to discover other connections that together with what you have lead to a solution!