KOMAL RANAWAT

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SELF LEARNING MATERIAL

April 5

2015LINES AND ANGLES

LINES AND ANGLES

Basic Terms and Definitions

Line

A line is a collection of points along a straight path. A line extends in both the directions and has no endpoints. It has no definite length.

Line segment

A line segment is a part of a line with two end points. There is only one line segment joining two given points.

Ray

A part of line with one end point is called a ray.

Collinear points

All the points that lie on the same line are called collinear points.

Angle

Two rays originating from the same end point form an angle.

Consider the two rays AB and AC originating from the same point A form an angle named as ∠BAC or ∠CAB or ∠A. The rays that form the angle are called the arms of the angle. The end point is called the vertex of the angle. The size of an angle is measured in degrees.

Types of angles

Acute angle

Angles measuring less than 90° but more than 0° are called acute angles.

Right angle

An angle measuring 90° is called a right angle. A right angle is formed by the intersection of two perpendicular lines.

Obtuse angle

Angles measuring greater than 90° but less than 180° are called obtuse angles.

Straight angle

An angle measuring 180° is called a straight angle.

Reflex angle

Angles measuring greater than 180° but less than 360° are called reflex angles.

Adjacent angles

Two angles are said to be adjacent if they have a common arm and a common vertex.

Linear pair of angles

Two adjacent angles are said to form a linear pair if their sum is 180°. Non-common arms of the linear pair of angles form a straight line

Vertically opposite angles

When two lines intersect, four angles are formed. The angles that are opposite to each other are called vertically opposite angles. There are two pairs of vertically opposite angles.

Complementary angles

Two angles are said to be complementary, if their sum is 90°.

Supplementary angles

Two angles are said to be supplementary if their sum is 180°.

Complementary and supplementary angles may or may not be adjacent angles.

Intersecting lines

Two or more lines that meet at one point are called intersecting lines. When two lines intersect each other, then the vertically opposite angles are equal. Sum of all the angles formed at a point is 360°.

Parallel lines

Lines on the same plane that never intersect are called parallel lines.

Transversal

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

When a transversal intersects two lines, m and n, eight angles are formed, four angles at each point, P and Q respectively. These angles are identified by their positions.

• ∠1, ∠2, ∠7 and ∠8 are called exterior angles

• ∠3, ∠4, ∠5 and ∠6 are called interior angles

• ∠1 and ∠5, ∠2 and ∠6, ∠4 and ∠8, ∠3 and ∠7 are pairs of corresponding angles

• ∠1 and ∠7, ∠2 and ∠8 are pairs of alternate exterior angles

• ∠4 and ∠6, ∠3 and ∠5 are pairs of alternate interior angles

• ∠4 and ∠5, ∠3 and ∠6 are consecutive interior angles on the same side of the transversal

If a transversal intersects two parallel lines, then

• each pair of corresponding angles is equal.

• each pair of alternate interior angles is equal.

• each pair of interior angles on the same side of the transversal is supplementary

Two lines intersected by a transversal are parallel if, either

• any one pair of corresponding angles is equal, or

• any one pair of alternate interior angles is equal, or

• any one pair of interior angles on the same side of the transversal is supplementary

Lines which are parallel to the same line are parallel to each other.

The sum of three angles of a triangle is 180°.

If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

QUESTIONS

Q1. Prove that if two lines intersect ,then both pairs of vertically opposite angles are equal.

Q2. AB parallel to CD and EF parallel to DQ determine angle PDQ , angle AED and angle DEF.

Q3. An exterior angle of a triangle is 105° and its interior opposite angles are equal. Each of these equal angles is________.

Q4. 3 angles of a triangle are a, b & c. if c = a+b/2, then find value of 'c'