Some definitions are important for angle measurements such as right, obtuse, and acute angles. In this post we are going to talk about these concepts.

Adjacent supplementary angle

Obra de Euclides.

Definition: Given a ray $\overrightarrow{oa}$ and an opposite ray $\overrightarrow{oc}$. We project a ray $\overrightarrow{ob}$ at a point in the plane. We say that the angle formed by points $b\hat{o}c$ is supplementary (supplement) and adjacent (neighbor) to the angle formed by points $a\hat{o}b$.

Adjacent supplementary angle

Straight angle

Obra de Euclides.

Definition (Right Angle): is every angle congruent to its adjacent supplementary.

Right Angle

Acute angle

Obra de Euclides.

Definition (Acute Angle): is every angle smaller than the right angle.

Acute angle

Obtuse angle

Obra de Euclides.

Definition (Obtuse angle): is every angle greater than the right angle.

Obtuse Angle

Measure Angles or Amplitude

Definition (Angle Measures): the measure of an angle is an ositive real number, associated with the angle. If $a\hat{o}b$ is any angle, the measure of this angle is given by $m(a\hat{o})b$

They satisfy some properties:

  • Angles are congruent if and only if they have equal measures.
$$a\hat{o}b \equiv c\hat{p}d \quad \Leftrightarrow \quad m(a\hat{o}b) = m(c\hat{p}d)$$
  • An angle is greater than the other if and only if the measure of one is greater than the other. It goes for minors too.
$$a\hat{o}b > c\hat{p}d \quad \Leftrightarrow \quad m(a\hat{o}b) > m(c\hat{p}d)$$
  • The sum of the measures of angles is worth.
$$rt \equiv a\hat{o}b + c\hat{p}d \quad \Rightarrow \quad m(rt) = m(a\hat{o}b) + m(c\hat{p} d)$$

Angle measurement units

Obra de Euclides.

Angle of one degree $(1^\circ)$ is the submultiple of 90 (ninety) of a right angle.

$$\text{one degree angle} = \dfrac{\text{right angle}}{90}$$
  • a right angle has 90 degrees $(90^\circ)$
  • an acute angle is less than 90 degrees $(90^\circ)$
  • an obtuse angle is greater than 90 degrees $(90^\circ)$

One minute angle

Obra de Euclides.

One minute angle $(1^\prime)$ is the submultiple of 60 (sixty) of a one degree angle.

$$1^\prime = \dfrac{1^\circ}{60}$$
  • a degree has 60 minutes $(60^\prime)$

Angle of a second

Angle of one second $(1^{\prime\prime})$ is the submultiple of 60 (sixty) of an angle of one minute.

$$1^{\prime\prime} = \dfrac{1^\prime}{60}$$
  • a minute has 60 seconds $(60^{\prime\prime})$
Obra de Euclides.

These are concepts of angle measurements. Wait for the next posts with consequences of this matter.