Angle comparisons are important for properties that will be used in later posts.

Congruence

Congruence in angles is a concept that is similar to equality of numbers. Here congruence is used with the symbol ($\equiv$) and this primitive notion satisfies the postulates:

  • Reflective
    • every angle is congruent to itself.
    • $\widehat{ab} \equiv \widehat{ab}$
  • Symmetric
    • If $\widehat{ab} \equiv \widehat{cd}$ then $\widehat{cd} \equiv \widehat{ab}$
  • Transitive
    • If $\widehat{ab} \equiv \widehat{cd}$ and $\widehat{cd} \equiv \widehat{ef}$ then $\widehat{ab} \equiv \widehat{ef}$

Angle Congruence

Angle Transport

Angle transport postulate: given an angle $a\hat{o}b$ and a half-line $\overrightarrow{oe}$ of a plane, exists on this plane and one of the half-planes that $\ overrightarrow{oe}$ allows to determine a single straight line $\overrightarrow{of}$ that forms with $\overrightarrow{oe}$ an angle that is congruent to the angle $a\hat{o}b$

Angle Transport

Comparison of angles

In comparing angles, we will use the angle transport postulate;

  • Angle $\widehat{ab}$ greater than $\widehat{cd}$

Bigger Angle

  • Angle $\widehat{ab}$ congruent to $\widehat{cd}$

Equal Angle

  • Angle $\widehat{ab}$ less than $\widehat{cd}$

Smaller Angle

Add angles

If the ray $ob $ is internal to the angle $a\hat{o}c$, the angle $a\hat{o}c$ is the sum of the angles $a\hat{o}b$ and $b \hat{o}c$.

Sum of Angles

$$\widehat{ac} = \widehat{ab} + \widehat{bc}$$

Sum of Angles

Angle Bisector

Definition: the bisector of an angle is a ray, which starts from the vertex of the angle and divides it into two congruent angles ($\cong$).

Drunk Walk.

From the figure, we notice that

  • $\angle \widehat{aob} \cong \angle\widehat{boc}$
  • Angle $\widehat{aob}$ is congruent to angle $\widehat{boc}$.
  • The ray b is the bisector of this angle.

These are concepts of comparing angles. Wait for the next posts with consequences of this matter.