Angle and Angle Bisector
Angle and Angle Bisector
Angle: The corners made by the intersection of two lines or line segments are called Angles.
Angle Bisector: A ray AX is said to be the bisector of , if X is a point in the interior of
, and
Thus, if ray AX is the bisector of , then
Theorem: 1 If two parallel lines are intersected by a transversal, the bisectors of any pair of alternate interior angles are parallel.
Converse of theorem: If the bisectors of a pair of alternate angles formed by a transversal with two given lines are parallel, then the given lines are parallel.
Theorem 2: If two parallel lines are intersected by a transversal, then bisectors of any two corresponding angles are parallel.
Converse of theorem: If the bisectors of any pair of corresponding angles formed by a transversal with two given lines are parallel, then the given lines are parallel.
Theorem 3: If the bisectors of angles of a triangle ABC meet at a point O, then
Proof:
Given: A triangle ABC such that the bisectors of meet at a point O.
To Prove:
Proof: In BOC, we have
In triangle ABC
................(i)
[
are bisectors of
and
respectively.
]
[dividing both sides by 2]
................(ii)
Substituting the value from equation (ii) in equation (i), we get
Hence the proof.
Theorem 4: If two parallel lines are intersected by a transversal, then the bisectors of the two pairs of interior angles enclose a rectangle.
Theorem 5: The sides AB and AC of a are produced to P and Q respectively. If the bisectors of
intersects at O, then
TQ and TR are the bisectors of
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