Angle Bisector Theorem

An angle bisector cut an angle precisely in half. One essential property of angle bisectors is that if a point is top top the bisector of one angle, climate the point is equidistant from the political parties of the angle. This is referred to as the Angle Bisector Theorem.

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In other words, if \(\overrightarrowBD\) bisects \(\angle ABC\), \(\overrightarrowBA\perp FD\overlineAB\), and, \(\overrightarrowBC\perp \overlineDG\) then \(FD=DG\).

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Figure \(\PageIndex1\)

The converse that this organize is also true.

Angle Bisector organize Converse: If a point is in the inner of one angle and equidistant native the sides, climate it lies on the bisector of that angle.

When we construct angle bisectors because that the angle of a triangle, they fulfill in one point. This suggest is referred to as the incenter the the triangle.

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Solution

No since \(B\) is no necessarily equidistant indigenous \(\overlineAC\) and also \(\overlineAD\). We execute not understand if the angles in the diagram are right angles.


Example \(\PageIndex2\)

A \(108^\circ\) edge is bisected. What space the measures of the result angles?

Solution

We recognize that to bisect way to cut in half, so every of the resulting angles will be fifty percent of 108. The measure up of each resulting edge is \(54^\circ\).


Example \(\PageIndex3\)

Is \(Y\) top top the edge bisector of \(\angle XWZ\)?

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Example \(\PageIndex5\)

\(\overrightarrowAB\) is the angle bisector the \(\angle CAD\). Fix for the missing variable.

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