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Mathematics LibreTexts

5.1: Right, Acute and Obtuse Angles

  • Page ID
    23605
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    • If \(|\measuredangle AOB| = \dfrac{\pi}{2}\), we say that \(\angle AOB\) is right;
    • If \(|\measuredangle AOB| < \dfrac{\pi}{2}\), we say that \(\angle AOB\) is acute;
    • If \(|\measuredangle AOB| > \dfrac{\pi}{2}\), we say that \(\angle AOB\) is obtuse.

    截屏2021-02-03 下午4.26.12.png

    On the diagrams, the right angles will be marked with a little square, as shown.

    If \(\angle AOB\) is right, we say also that \([OA)\) is perpendicular to \([OB)\); it will be written as \([OA) \perp [OB)\). From Theorem 2.4.1, it follows that two lines \((OA)\) and \((OB)\) are appropriately called perpendicular, if \([OA) \perp [OB)\). In this case we also write \((OA) \perp (OB)\).

    Exercise \(\PageIndex{1}\)

    Assume point \(O\) lies between \(A\) and \(B\) and \(X \ne O\). Show that \(\angle XOA\) is acute if and only if \(\angle XOB\) is obtuse.

    Hint

    By Axiom IIIb and Theorem 2.4.1, we have \(\measuredangle XOA - \measuredangle XOB \equiv \pi\). Since \(|\measuredangle XOA|, |\measuredangle XOB| \le \pi\), we get that \(|\measuredangle XOA| + |\measuredangle XOB| = \pi\). Hence the statement follows.