Introduction to divide an angle into two equal angles:
Let us assume an angle formed by two rays. The vertex of the angle is the initial points for both the rays. Suppose you need to divide this angle into two equal angles, what is to be done.
One method is to take the help of a protractor to measure the angle. Then you numerically divide that by two, mark a point again using the protractor and join that to the vertex of the angle.
However the better method to divide an angle into two equal angles is by using a compass.
Divide an Angle into Two Equal Angles – Compass Method
The following diagram illustrates the method to divide an angle into two equal angles.
Let OX and OY be the rays making an angle XOY at vertex O.
Set the compass to a convenient radius and strike two arcs from O, one on the ray OX and another on the ray OY. Let the points of intersection of these arcs be A and B respectively. (Fig i)
Now set the compass again for a convenient radius, draw an arc from point A. Without changing the compass setting, draw a similar arc from point B. Let these arcs intersect at P. Draw a ray connecting the vertex O and the point P. The ray OP divides angle XOY into two equal angles. (Fig ii)
The ray OP is called the internal angle bisector of the angle XOY.
I am planning to write more post on cbse question papers for class 12th. Keep checking my blog.
Divide an Angle into Two Equal Angles – Proof for the Method
Join AP and BP (Fig iii)
Considering the triangles OAO and OBP,
OA = OB (By construction)
AP = BP (By construction)
Side OP is common.
Hence the triangles OAO and OBP are congruent as per SSS postulate.
Therefore , angle AOP = angle BOP (corresponding parts of congruent triangles)
Thus it is proved that the angle XOY is divided into two equal angles.
Let us assume an angle formed by two rays. The vertex of the angle is the initial points for both the rays. Suppose you need to divide this angle into two equal angles, what is to be done.
One method is to take the help of a protractor to measure the angle. Then you numerically divide that by two, mark a point again using the protractor and join that to the vertex of the angle.
However the better method to divide an angle into two equal angles is by using a compass.
Divide an Angle into Two Equal Angles – Compass Method
The following diagram illustrates the method to divide an angle into two equal angles.
Let OX and OY be the rays making an angle XOY at vertex O.
Set the compass to a convenient radius and strike two arcs from O, one on the ray OX and another on the ray OY. Let the points of intersection of these arcs be A and B respectively. (Fig i)
Now set the compass again for a convenient radius, draw an arc from point A. Without changing the compass setting, draw a similar arc from point B. Let these arcs intersect at P. Draw a ray connecting the vertex O and the point P. The ray OP divides angle XOY into two equal angles. (Fig ii)
The ray OP is called the internal angle bisector of the angle XOY.
I am planning to write more post on cbse question papers for class 12th. Keep checking my blog.
Divide an Angle into Two Equal Angles – Proof for the Method
Join AP and BP (Fig iii)
Considering the triangles OAO and OBP,
OA = OB (By construction)
AP = BP (By construction)
Side OP is common.
Hence the triangles OAO and OBP are congruent as per SSS postulate.
Therefore , angle AOP = angle BOP (corresponding parts of congruent triangles)
Thus it is proved that the angle XOY is divided into two equal angles.
