2.5: Ángulos verticales

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Un par de ángulos$AOB$ y$A'OB'$ se llama vertical si el punto$O$ se encuentra entre$A$$A'$ y$B$ y entre y$B'$ al mismo tiempo.

Proposición$\PageIndex{1}$

Los ángulos verticales tienen medidas iguales.

Prueba

Supongamos que los ángulos$AOB$ y$A'OB'$ son verticales. Tenga en cuenta que$\angle AOA'$ y$\angle BOB'$ son rectos. Por lo tanto,$\measuredangle AOA' = \measuredangle BOB' = \pi$.

$2021-02-02 10.35.53.png$

De ello se deduce que

\[\begin{array} {rcl} {0} & = & {\measuredangle AOA' - \measuredangle BOB' \equiv} \\ {} & equiv & {\measuredangle AOB + \measuredangle BOA' - \measuredangle BOA' - \measuredangle A'OB' \equiv} \\ {} & \equiv & {\measuredangle AOB - \measuredangle A'OB'.} \end{array}\]

Desde$-\pi < \measuredangle AOB \le \pi$ y$-\pi < \measuredangle A'OB' \le \pi$, lo conseguimos$\measuredangle AOB = \measuredangle A'OB'$.

Ejercicio$\PageIndex{1}$

Supongamos que$O$ es el punto medio para ambos segmentos$[AB]$ y$[CD]$. $AC = BD$Demuéstralo.

Pista: Aplicando la Proposición 2.5.1, lo conseguimos$\measuredangle AOC = \measuredangle BOD$. Queda por aplicar el Axioma IV.

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