Angles are two semi-straight lines that have the same origin, at the vertex, and are measured in degree (º) or in radian (rad), according to the International System.
Types of Angles
Depending on your measurements, the angles are classified as acute, straight, obtuse and shallow.
Acute
The acute angle measures less than 90º (
Straight
The right angle measures the same as 90º (= 90º).
Obtuse
The obtuse angle measures more than 90º and less than 180º (90º>
Shallow
The shallow angle, also known as a half turn, measures the same as 180º (= 180º).
How to measure angles?
To measure the angles, we need a protractor, an instrument in a circle (360º) or semicircle (180º) that is divided into degrees, and follow the following steps:
- Place the center of the protractor base on the apex of the angle.
- Place the point that indicates 0º of the protractor on one side of the angle.
- The other side of the angle will point to your measurement.
The angle is the most used unit of measurement. Minute and second are its multiples.
It should be noted that 360º is equivalent to 2 π rad. Thus, 180º is equivalent to π rad.
Complementary Angles
Complementary angles are those that together measure 90º.
30º + 60º = 90º, which means that the angles complement each other, 30º complements the angle of 60º and vice versa.
Supplementary Angles
Supplementary angles are those that together measure 180º.
135º + 45º = 180º
This means that the angle of 135º is the supplement of the angle that measures 45º.
At the same time, the 45º angle is the supplement of the 135º angle.
Adjacent Angles
The adjacent angles, which are those that have no common points, can be complementary or supplementary.
The sum of the complementary adjacent angles is 90º.
The sum of the supplementary adjacent angles is 180º.
Compare the difference between adjacent angles with other angles that have internal points in common.
AÔC and AÔB have internal points in common. Therefore, they are not adjacent.
AÔC and CÔB do not have internal points in common. Therefore, they are complementary adjacent .
AÔB and AÔC do not have internal points in common. Therefore, they are supplementary adjacent .
Congruent Angles
Congruent angles are those that have the same measure.
Vertex Opposite Angles:
Angles opposed by the vertex (OPV) are those whose sides are opposed to the sides of another angle.
Exercises
1. (MACKENZIE-2014) In the figure below, a and b are parallel lines.
The correct statement regarding the number that expresses, in degrees, the measure of the angle is:
a) a prime number greater than 23.
b) an odd number.
c) a multiple of 4.
d) a divisor of 60.
e) a common multiple between 5 and 7.
Answer
Alternative d: a divisor of 60.
2. (IFPE-2012). Júlia started studying geometry at her school. Doubtfully in an exercise by the math teacher, she asked her uncle for help.
The statement was: ‘The straight lines are parallel; the uet lines, two transversal. Find the value of angle x in the figure below ‘. Therefore, the value of x is:
a) 120 °
b) 125 °
c) 130 °
d) 135 °
e) 140 °
Answer:
Alternative e: 140 °.