Category : 9th Class
Lines and Angles
The angle formed by the two rays \[\overline{AB}\,\,and\,\,\overline{AC}\text{ }is\text{ }\angle BAC\text{ }or\text{ }\angle CAB.~\]called \[\overline{AB}\,\,and\,\,\overline{AC}\]are called the arms and the common initial point ‘A’ is called the vertex of the angle.
e.g., In the adjacent figure, the line OP divides \[\angle \]AOB into two
Equal parts.
\[\angle AOP=\angle POB={{\operatorname{x}}^{o}}\]
So, the line OP is ‘called the bisector of \[\angle \]AOB.
(i) Complementary angles: Two angles are said to be Complementary if the sum of their measures is equal to\[~{{90}^{o}}\]
Here\[\angle x+\angle y={{90}^{o}},\]therefore \[\angle x\,\operatorname{and}\,\angle y\] Complementary angles:
(ii) Supplementary angles: Two angles are said to be supplementary if the sum of their measures isequal to\[{{180}^{o}}\].
Here \[\angle x+\angle y={{180}^{o}},\]therefore \[\angle x\,\operatorname{and}\,\angle y\]
Supplementary angles.
Note: \[\angle AOC=\angle AOB+\angle BOC\]
Moreover,\[\angle \text{ }BAC\text{ }+\text{ }\angle DAC={{180}^{o}}\]
Note: Linear pair of angles \[\xrightarrow[\mathbf{always}]{\mathbf{are}}\]Adjacent angles.
Adjacent angles \[\xrightarrow[\mathbf{always}]{\mathbf{are}\,\,\mathbf{not}}\] Linear pair angles.
(ii) Linear pair of angles are supplementary.
In the above figure,\[\angle BOD\text{ }and\text{ }\angle AOC\] is a pair of vertically opposite angles because they have a common vertex and also OB, OA; OC, OD are two pairs of opposite rays.
Similarly, we find that \[\angle BOD\text{ }and\text{ }\angle AOC\]is a pair of vertically opposite angles because they have common vertex at 0 and also OB, OA; OC; OD are two pairs of opposite rays.
Note: If two lines intersect each other’s, the vertically opposite angles are equal.
Transversal: A line intersecting two or more distinct lines at distinct points is called a transversal.
In the adjacent figure, transversal EF intersects lines AB and CD at points P and Q respectively.
the angles are not adjacent angles. In the above figure, the pairs of corresponding angles are:
(iii)\[\angle \]3 and\[\angle \]7 and (iv) \[\angle \]4 and \[\angle \]8
said to beco-ihterior angles if they are interior angles and lie on the same side of the
transversal. In the above figure, the pairs of co-interior angles are:
(i) \[\angle \]3 and \[\angle \]6 and (ii)\[\angle \]4 and \[\angle \]5
(i) \[\angle \]4 and \[\angle \]6 and (ii) \[\angle \]3 and \[\angle \]5
(i) If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
(ii) Converse: If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal (i.e., co-interior angles) is supplementary.
The sum of the three angles of a triangle is 180°.
In AABC, ZA + ZB + ZC = 180°.
In\[\Delta ABC,\text{ }\angle ACD\text{ }=\angle BAC+\angle ABC\].
Note: An exterior angle of a triangle is greater than each of its interior angles.
In the above figure, \[\angle \]ACD is greater than \[\angle \]BAC as well as \[\angle \]ABC.
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