7th Class Mathematics Lines and Angles

Lines and Angles

Category : 7th Class

Lines and Angles

Point: A point is a geometrical representation of a location. It is represented by a dot.

Line: A geometrical line is a set of points that extends endlessly in both the directions i.e., a line has no end points. A line AB is represented as.\[\overleftrightarrow{AB}\]

Line segment: A line segment is a part of a line. A line segment has two end points. A line segment AB is represented as\[\overline{AB}\].

Ray: A ray is a part of the line which has one end point (namely its starting point).

A ray OP is denoted as\[\overrightarrow{OP}\].

Angle: An angle is the union of two rays with a common initial point.

The symbol of angle is\[\angle \]. An angle is measured in degrees\[\left( {}^\circ \right)\].

The angle formed by the two rays \[\overrightarrow{AB\,}\,and\,\overrightarrow{AC}\]is denoted by \[\angle \]BAC or \[\angle \]CAB

The two rays \[\overrightarrow{AB\,}\,and\,\overrightarrow{AC}\] are called the arms and the common initial point 'A' is called the vertex of the angle ABC.

Types of Angles:

(i) Right angle: An angle whose measure is equal to \[{{90}^{o}}\]is called a right angle.

(ii) Acute angle: An angle whose measure is less than 90° is called an acute angle.

(iii) Obtuse angle: An angle whose measure is greater than 90° but less than 180° is called an obtuse angle.

(iv) Straight angle: An angle whose measure is equal to \[{{180}^{o}}\] is called a straight angle.

(v) Complete angle: An angle whose measure is exactly equal to 360° is called a complete angle.

Reflex angle: An angle which is greater than 180° but less than \[{{360}^{o}}\] is called a reflex angle.

Zero angle: An angle whose measure is 0° is called a zero angle.

Related Angles:

(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°, therefore \[\angle \]x and \[\angle \]y are complementary angles.

(iii) Supplementary angles: Two angles are said to be supplementary if the sum of their measures is equal to\[{{180}^{o}}\].

Here, \[\angle \]x + \[\angle \]y = 180°, therefore \[\angle \]x and \[\angle \]y are supplementary angles.

Adjacent angles: Angles having a common vertex, a common arm and the non-common arms lying on either side of the common arm are called adjacent angles.

In the given figure,\[\angle \]AOB and\[\angle \]COB have a common vertex'0', a common arm \[\overrightarrow{OB}\text{ }and\text{ }\overrightarrow{OA}\text{ }and\text{ }\overrightarrow{OC}\]are on opposite sides of \[\overrightarrow{OB}\]. So they are adjacent angles.

Linear pair of angles: Two adjacent angles make a linear pair of angles, if the non-common arms of these angles form two opposite rays (with same end point).

In the figure given, the angles BAC and CAD form a linear pair of angles because the non - common arms AB and AD of the two angles are the opposite rays, with the same vertex A.

Moreover,\[\angle \]BAC +\[\angle \]DAC = \[{{180}^{o}}\] .

Note: 1. A liner pair is always supplementary.

A liner pair is always adjacent need not be a linear pair.

Vertically opposite angles: Two angles having the same vertex are said to form a pair of vertically opposite angles, if their arms form two pairs of opposite rays.

In the figure given, \[\angle \]BOD and \[\angle \]AOC are 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. Vertically opposite angles are formed when two lines intersect.

Similarly, we find that\[\angle \]BOC and \[\angle \]AOD is another pair of vertically opposite angles. Pair of Lines:

Note: If two lines intersect each other, the vertically opposite angles formed are equal.

Pair of lines:

Intersecting lines: Two lines which are distinct and have a common point are called intersecting lines. The common point is called the point of intersection of the two lines.

Perpendicular lines: If two lines \[l\] and m intersect at right angles, they are called perpendicular lines, denoted as \[l\]\[\bot \]m, read as \[l\] is perpendicular to m.

Parallel lines: Two lines\[l\] and m are said to be parallel, if they lie in the same plane and do not intersect when produced however far on either side and is written as\[l\]\[\parallel \]m read as\[l\]is parallel to m.

Transversal: A line which intersects two or more lines at distinct points is called a transversal.

In the given figure, p is a transversal to the lines I and m.

Angles made by a transversal:

In the figure given, lines I and m are cut by the transversal p. The eight angles marked 1 to 8 have names given in the table.

Interior angles	\[\angle \]3,\[\angle \]4,\[\angle \]5,\[\angle \]6
Exterior angles	\[\angle \]1,\[\angle \]2,\[\angle \]7,\[\angle \]8
Pairs of Corresponding angles	\[\angle \]1 and \[\angle \]5,\[\angle \]2 and \[\angle \]6,\[\angle \]4 and \[\angle \]8,\[\angle \]3,\[\angle \]6
Pairs of alternate exterior angles	\[\angle \]1 and\[\angle \]7,\[\angle \]2 and \[\angle \]8
Pairs of interior angles on the same side of the transversal	\[\angle \]4 and \[\angle \]5,\[\angle \]3, and \[\angle \]5