Lines and Angles

Notes of chapter: Lines and Angles are presented below. Indepth notes along with worksheets and NCERT Solutions.

(1)Ray-

A line that has one end point and it extends infinitely in other direction, is called ray.

 

Ray of NCERT Chapter Lines and Angles

In the above diagram, PQ is a ray with end P.

(2) Line segment-

A part of a line that has two end points is called line segment.

 

Line Segment of NCERT Chapter Lines and Angles

In above diagram, PQ is a line segment with ends P and Q.

(3) Lne-

A line segment when extends its both end points in either direction endlessly, is called a line.

 

Line of NCERT Chapter Lines and Angles

In the above diagram, PQ is a line with no ends.

(4) Vertex-

The meeting point or common point of two rays or two line segments or two lines is called vertex.

 

Vertex of NCERT Chapter Lines and Angles

(5) Angle-

The distance between two rays or two line segments or two lines diverging from vertex (common point) is called an angle.

 

Angle of NCERT Chapter Lines and Angles

(6) Types of angles

 (i) Complementary angles-

When the sum of the two angles is 90o, the angles are complementary angles.

 

Complementary Angles of NCERT Chapter Lines and Angles

(ii)Supplementary angles.

When the sum of the two angles is 180o, the angles are supplementary angles.

 

Supplementary Angles of NCERT Chapter Lines and Angles

(iii) Adjacent angles.

When two angles have a common vertex, a common arm and non – common arms are on either side of the common arm, are called adjacent angles.

 

Adjacent angles with common arm and common vertex of NCERT Chapter Lines and Angles

(iv) Linear pair-

A linear pair is a pair of adjacent angles whose non-common sides are opposite rays.

 

Linear Pair of NCERT Chapter Lines and Angles

(v)Vertically opposite angles

When two lines intersect, the vertically opposite angles so formed are equal.

 

Vertically Opposite Angles of NCERT Chapter Lines and Angles

Definition of vertically opposite angles,lines and angles,NCERT

(vi) Acute angle-

An acute angle is an angle which measures between 00 and 900.

 

Acute Angle of NCERT Chapter Lines and Angles

(vii) Obtuse angle-

An obtuse angle is an angle which measures greater than 900 and less than 1800.

 

Obtuse Angle of NCERT Chapter Lines and Angles

(viii) Right angle-

A right angle is an angle which measures equal to 900.

 

Right Angle of NCERT Chapter Lines and Angles

(ix) Straight angle

A straight angle is an angle that measures equal to 1800.

Straight Angle of NCERT Chapter Lines and Angles

(x)Reflex angle-

A reflex angle is an angle which measures greater than 1800 but less than 3600.

 

Reflex Angle of NCERT Chapter Lines and Angles

(7)Pairs of Lines

(i)When two or more lines meet at a common point are called intersecting lines and this common point is called point of intersection.

Intersecting lines and point of intersection of NCERT Chapter Lines and Angles

In the above figure, l and m are intersecting lines and point O is point of intersection.

(ii)Transversal line-

A line that intersects two or more lines at distinct points is called a transversal line.

 

Transversal Lines of NCERT Chapter Lines and Angles

In the above figure, p is transversal to the lines l, m and n.

(iii) Angles made by a transversal

(a)Interior angles-

When two lines are cut by a third line (transversal), then the angles formed inside the lines are called interior angles.

 

Interior Angles of NCERT Chapter Lines and Angles

Definition of interior angle ,lines and angles,NCERT

(b)Exterior angle-

When two lines are cut by a third line (transversal), then the angles formed outside the lines are called exterior angle.

 

Exterior Angles of NCERT Chapter Lines and Angles

(c) Corresponding angles-

The angles which are on the same side of the transversal and have different vertices are called corresponding angles. They are in ’corresponding’ position (above or below, left or right) relative to the two lines.

Corresponding Angles of NCERT Chapter Lines and Angles

(d) Alternate interior angles-

The angles which are on opposite sides of the transversal and have different vertices are called alternate interior angles. These angles lie ‘between’ the two lines.

 

Alternate Interior Angles of NCERT Chapter Lines and Angles

(e) Alternate exterior angles-

The angles which are on opposite sides of the transversal and have different vertices are called alternate exterior angles. These angles lie outside the two lines.

(iv)Transversal of parallel lines

A line that intersects two or more parallel lines at distinct points is called a transversal of parallel line.

Transversal Parallel Lines of NCERT Chapter Lines and Angles

In the above figure, l is a transversal line and m & n are parallel lines.

Angles made by Transversal of Parallel lines

(a)If two parallel lines cut by a transversal, each pair of corresponding angles is equal in measure.

(b) If two parallel lines cut by a transversal, each pair of alternate interior angles is equal in measure.

(c) If two parallel lines cut by a transversal, each pair of interior angles on the same side of the transversal is supplementary.

 

(8)Checking for parallel lines

(i) When, a transversal cuts two lines, such that pairs of corresponding angles are equal, then the lines have to be parallel.

(ii) When, a transversal cuts two lines, such that pairs of alternate interior angles are equal, then the lines have to be parallel.

(iii) When, a transversal cuts two lines, such that pairs of interior angles on the same side of the transversal are supplementary, then the lines have to be parallel.

(9) Pairs of Angles

Axiom 1- If a ray stands on a line, then the sum of two adjacent angles so formed is 1800.

Given – Line AB and a ray OC.

Construction – Draw a perpendicular DO at point O.

Linear Pair Angles Axiom of NCERT Chapter Lines and Angles Linear Pair Angles Axiom of NCERT Chapter Lines and Angle

Theorem: Vertical Opposite Angles of NCERT Chapter Lines and Angles

Corresponding Angles Axiom of NCERT Chapter Lines and Angles

Converse of Corresponding Angles Axiom NCERT Chapter Lines and Angles

(11) Lines parallel to the same line

If two lines are parallel to the same line, they are parallel to each other.

Given – Line m  line l and line n  line l.

Construction – Draw a transversal line t which intersects line l, m and n.

 

Lines Parallel to Same Line of NCERT Chapter Lines and Angles

Angle Sum Property of a Triangle of NCERT Chapter Lines and Angles

Exterior Angle is Equal to Sum of its Opposite Interior Angles of NCERT Chapter Lines and Angles

Helping Topics

NCERT Solutions Class 7

NCERT Solutions Class 9

Worksheet Class 7

Worksheet Class 9

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