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Maths / Constructions / Construction of Angle and Angle Bisector

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Construction of Angle and Angle Bisector

Bisecting an angle means drawing a ray in the interior of the angle, with its initial point at the vertex of the angle such that it divides the angle into two equal parts

In order to draw a ray AX bisecting a given angle $large angle BAC,$ we follow the following steps.
STEP I With centre A and any convenient radius draw an are cutting AB and AC at P and Q respectively.
STEP II With centre P and radius more than half of PQ draw an arc.
STEP III With centre Q and the same radius,as in step II, draw another arc intersecting the arc in step II at R.
STEP IV Join AR and produce it to any point X. The ray AX is the required bisector of $large angle BAC$ .

Verification: Measure $large angle BAX$ and $large angle CAX.$ You would find that $large angle BAX=angle CAX$

Justification: Now let us see how this method gives us the required angle bisector.

Join PR and QR.

In $Delta$ APR and $Delta$ AQR, we have

AP = AQ [ $large because$ AP and AQ are radii of the same arc]

PR = QR [ $large because$ PR and QR are arcs of equal radii.]

AR = AR [common] $large 72^{circ}$

So, $Delta$ APR $cong$ $Delta$ AQR [by SSS congruence criterion ]

$large Rightarrow$ $large angle PAR=angle QAR$ [C.P.C.T.]

Hence, AR is the bisector of $large angle BAC.$

ILLUSTRATION: Using a protractor, draw an angle of mesure $large 72^{circ}$ . With this angle as given, draw an angle of measure $large 36^{circ}$ .

SOLUTION We follow the following steps to draw an angle of $large 36^{circ}$ from an angle of .

Steps of Construction

STEP I Draw a ray OA as shown in fig. STEP II With the help of a protractor contruct an angle AOB of measure $large 72^{circ}$ .
STEP III WIth centre O and a convenient radius draw an arc cutting sides OA and OB at P and Q respectively.
STEP IV With centre P and radius more than half of PQ, draw an arc.
STEP V With centre Q and the same radius, as in the previous step, draw another arc intersecting the arc drawn in the previous step at R.
STEP VI Join OR and produce it to form ray OX.
The angle $large angle AOX$ so obtained the required angle of measure $large 36^{circ}$

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Construction of Angle and Angle Bisector

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