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Definition and types of angles

Draw two rays starting from the same point. These two rays divide the plane into two regions.

interior
region
(angle)

exterior
region
(angle)

Two rays originating from the same point divide the plane into two angles. The rays are the arms of the angle, and the starting point is the vertex.

arm

angle

arm

P

vertex

Since rays form two angles, you must indicate which angle you refer to (see image below).

Angles can be named using the lowercase letters of the Greek alphabet (e.g. α) or using a point on one of the arms, the vertex and a point on the other arm (e.g. BCA∢). Note that the vertex must always be in the centre.

C

A

B

α

γ

β

C

B

A

α = BAC∢ = CAB∢

BCA∢ or ACB∢

The angle of two intersecting lines

Two intersecting lines divide the plane into four regions.
The following statements are valid for the resulting angles: α = γ and β = δ.

When referring to the angle formed by intersecting lines, you must think of the smaller angle.

α

γ

β

δ

e

f

Types of angles

a) If the arms of an angle are perpendicular, the smaller angle formed by them is called a right angle. Symbol: ⦝ or ⦜.

b) If the two arms form a straight line, the angle formed by them is called a straight angle.

c) If the two arms are on top of each other, the larger angle formed by them is called a full angle, while the smaller one is called a zero angle.

zero angle

acute angle

right angle

obtuse angle

straight angle

reflex angle

full angle

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zero angle

acute angle

right angle

obtuse angle

straight angle

reflex angle

full angle

Acute angles are larger than zero angles but smaller than right angles.
Obtuse angles are larger than right angles but smaller than straight angles.

Reflex angles are larger than straight angles but smaller than full angles.
Reflex angles are concave, while the rest are convex.

Exercises

{{exercise_number}}. Fold a sheet of paper in half, then fold it in half again. Finally, unfold the paper sheet.

What is the position of the folds compared to each other, and what type of angles do they form? (→)

{{exercise_number}}. Draw triangles that meet the following criteria.

a) All of its angles are acute angles.

b) It has a right angle.

c) It has an obtuse angle.

d) It has two right angles.

{{exercise_number}}. Draw quadrilaterals that meet the following criteria.

a) All of its angles are right angles.

b) It has a right angle.

c) It has exactly one right angle.

d) It has an obtuse angle.

e) All of its angles are acute angles.

f) It has exactly two right angles.

{{exercise_number}}. A boat heads north, then takes a right-angle turn.

In which direction does it sail after the turn?

{{exercise_number}}. Make two discs with the same radius using paper sheets of different colours (see image). Cut both discs along a radius, then put them on top of each other. Start rotating one of the discs to form different types of angles. (→)

{{exercise_number}}. What types of angles are marked by the Greek letters?

α

β

γ

δ

ε

{{exercise_number}}. Write the angles marked by Greek letters using the vertices of the triangle and the quadrilateral. For example α = BAC∢ = CAB∢.

A

C

B

D

G

F

E

α

β

γ

δ

ω

ε

π

{{exercise_number}}. Draw the following angles in the image.

α = BAC∢

β = EFD∢

γ = GHI∢

δ = JKL∢

A

B

E

F

G

H

J

L

C

D

I

K

{{exercise_number}}. What types of angles are labelled on the letters?

α

β

α

β

δ

γ

{{exercise_number}}. What type of convex angle do the hands of a clock form at

a) \latex{ 12:00 };

b) \latex{ 12:30 };

c) \latex{ 15:00 };

d) \latex{ 17:00 };

e) \latex{ 17:30 };

f) \latex{ 18:00 };

g) \latex{ 19:30 };

h) \latex{ 21:00 };

i) \latex{ 21:45 ?}

{{exercise_number}}. What type of angle is it if it is

a) two times larger than an acute angle;

b) three times larger than an acute angle?

{{exercise_number}}. What type of angle is it if it is

a) half a reflex angle;

b) one-third of a reflex angle?

{{exercise_number}}. Draw a quadrilateral that has

a) a reflex angle;

b) a reflex and a right angle.

{{exercise_number}}.

a) What type of angles do cardinal and intercardinal directions form with the north?
b) List cardinal and intercardinal directions that are perpendicular.
c) List cardinal and intercardinal directions that form an acute angle.
d) List cardinal and intercardinal directions that form an obtuse angle.

Quiz

How many times do the hour and minute hands of a clock overlap during a day?

Mathematics 5.

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