Geometry for Elementary School/Angles
|The corresponding material in Euclid's elements can be found on page 26 of Book I in Issac Todhunter's 1872 translation, The Elements of Euclid for the Use of Schools and Colleges.|
In this section, we will talk about angles.
An angle (∠) is made up of a vertex (a point), two arms (rays), and an arc. They are arranged so that the endpoint of the arms are the same as the vertex, and the arc runs from one arm to another. The size of an angle depends on how big the arms are opened, and they are measured in degrees. You can measure them by putting your protractor on the vertex and looking at the degrees your second arm has reached.
An angle that is less than 90° is known as an acute angle. A 90° angle is known as a right angle. Those between 90° and 180° are obtuse angles. Exactly 180° angles are called straight angles. Those between 180° and 360° are reflex angles, while angles at 360° are round angles
An angle is usually named by the points it contains. The format is as follows:
"∠" + a point on one arm + vertex + a point on the other arm
However, sometimes there are no angles on that vertex, and we can omit the point on the arms. In fact, when we are lazy, we can even use a lowercase letter to represent a certain angle. Note that in this case, ∠ must be omitted. Although the lowercase letter represents the value of the angle, all of these names can be used as unknowns in equations.
Adjacent angles (adj. ∠s) are angles where:
- Their opposite arms coincide (overlap);
- Their arcs do not coincide (overlap);
- Their vertices coincide (overlap).
Sometimes, two angles may add up to 90° or 180°. They are called complementary angles and supplementary angles respectively. As many angles have such properties, these will be quite handy in the future.
Angles at a point
Sometimes, two or more angles share a common vertex, and their sizes add up to 360o. They are called angles at a point (∠s at a pt.). This can be very useful when we write proofs or find out angles.
For example, imagine that O is a point in the figure. The three points, A, B, and C are around the point O, and a ray shoots out of O to A, B and C respectively. Given that ∠AOB = 120° and ∠BOC = 150°,
Adjacent angles on a straight line
When the sizes of adjacent angles add up to 180°, they are adjacent angles on a straight line. They are used when finding out the value of one of the angles. (Or more, for that matter, when you have angles that are equal or related.) The abbreviation, adj. ∠s on st. line, can be used as a reference that the angles add up to 180°.
Look at the image on the right as an example. Here, b and a are supplementary. The sum of b and a is equal to c. b and a are adjacent angles on a straight line. If we know the value of b, we can find out the value of a easily. Note that a, b, and c are angles at a point.
Vertically opposite angles
Vertically opposite angles are very simple. If two straight lines run into each other, the opposite angles produced must be vertically opposite angles (vert. opp. ∠s). They must be equal to each other. Note that you cannot assume that something is a straight line just by observation, so be sure that it's mentioned in the question before you do anything. Vertically opposite angles is a very common reference and will come in handy in many situations, so before you are stuck on a problem, see if you can find some vertically opposite angles first.
Look at the figure on the right. As indicated in this figure, D is equal to C and A is equal to B. This is because they are vertically opposite angles. Note that here, D and A, A and C, C and B, and D and B are all pairs of adjacent angles on a straight line. Also, the four angles are angles at a point.