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Class IX - maths: Lines and Angles
One Word Answer Questions:
Q) What is Acute Angle?
A)  

Q) What is Right Angle?
A)  

Q) What is Obtuse Angle?
A)  

Q) What is Straight Angle?
A)  

Q) What is Reflex Angle?
A)  

Q) What is Zero Angle?
A)  

Q) What is Line?
A)  

Q) What is Ray?
A)  

Q) Two adjacent angles are said to form a linear pair if their sum is?
A)  

Q) Two adjacent angles are said to form a linear pair if their sum is?
A)  

Short Answer Questions:
Q) What is Point?
A)

Q) What is Supplementary Angles?
A)

Q) What is Supplementary Angles?Give one Example?
A)

Q) What is Bisecting an Angle?Give one Example?
A)

Q) What is Interior and Exterior of an Angle?Give one Example?
A)

Q) What is Adjacent Angles?Give one Example?
A)

Q) Explain about Supplementary angles and Vertically opposite angles?
A)

Q) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180 degrees and vice-versa. This property is called as the ?
A)

Q) When the sum of the measures of two angles is 90°, such angles are called ?
A)

Q) An angle whose measure is 180° is called a?
A)

Q) the point of intersection of any two angular bisectors of a triangle are called?
A)

Long Answer Questions:
Q) POQ is a straight line and OS stands on PQ. Find the value of x and the measure of POS, SOR and ROQ?
    
A)

Q) Explain about Properties of Parallel Lines and Properties Of Angles Associated with Parallel Lines?
A)

Q) Use your protractor to draw 60°. and . Use your protractor to draw 90°.Explain ?
A)

Q) Explain about Adjacent Angles and Supplementary Angles?
A)

Q) Explain about Complementary Angles and Right Angle?
A)

Q) Find the complement of the angle 2/3 of 90°?
A)

Q) One angle is equal to three times its supplement. The measure of the angle is?
A)

Q) AB and CD are two parallel lines. PQ cuts AB and CD at E and F respectively. EL is the bisects of ∠FEB. If ∠LEB = 35°, then ∠CFQ will be?
A)

Q) Two lines AB and CD intersects at O. If ∠AOC + ∠COB + ∠BOD = 270°, then ∠AOC =?
A)



Content
  • Angle
  • Types of Angles
  • Comparision of Angles
  • Interior and Exterior of an Angle
  • Lines of Symmetry
  • Lines of symmetry in letters of the English alphabet
  • Construction of Angles by using Compass
  • Measuring an Angle by a Protractor
  • Parallel Lines
  • Properties Of Angles Associated with Parallel Lines
  • Complementary and Supplementary Angles
  • Solved problems on complementary and supplementary angles
  • Parallel and Transversal Lines
  • Worked-out problems for solving parallel and transversal lines
Angle

» Angle is discussed here in details. In our daily life we come across different objects having two arms hinged at a point.
» For example, hands of a clock, two arms of a divider, and two sharp edges of a scissors are all hinged at a point and thus are inclined to each other.
» This inclination between two arms is known as angle.
» When two rays have a common end point they from an angle. In this figure two rays OA and OB have a common end point O.
» So they form an angle AOB. The common end point is called the Vertex and rays OA and OB are the arms of the angle.

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» In the above figure, AOB is angle whose two arms are the rays OB and OA. The symbol for an angle is `/_` and angle AOB can be written as `/_`AOB. Every angle has a measure.
» The unit of measurement of an angle is degree (°). In a figure the measure of an angle is generally written between the arms of the angle, close to the vertex.


Types of Angles

Types of angles are discussed here according to their degree measure:

Acute Angle
Right Angle
Obtuse Angle
Straight Angle
Reflex Angle
Zero Angle

1. Acute Angle

An angle whose measure is less than 90° is called an acute angle.

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» `/_`MON shown in adjoining figure is equal to 60°. So, `/_`MON is an acute angle.


2. Right Angle

An angle whose measure is 90° is called right angle.

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» In the above figure, `/_`AOB is a right angle. In this case, we say that the arms OA and OB are perpendicular to each.
» Therefore, `/_`AOB shown in adjoining figure is 90°.
So, `/_`AOB is a right angle.


3. Obtuse Angle

An angle whose measure is greater than 90° but less than 180° is called an obtuse angle.

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» `/_`DOQ shown in the above figure is an obtuse angle.


4. Straight Angle

An angle whose measure is 180° is called a straight angle.

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» `/_`XOY shown in the above figure is a straight angle.
A straight angle is equal to two right angles.


5. Reflex Angle

An angle whose measure is more than 180° but less than 360° is called a reflex angle.

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» `/_`AOB shown in the above figure is 210°.
So, `/_`AOB is a reflex angle.


6.Zero Angle

An angle measure 0° is called a zero angle.

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When two arms of an angle lie on each other, 0° angle is formed.


Comparision of Angles

"An angle whose degree measure is greater than the degree measure of another angle is a greater angle". Thus, we can say that:
"Acute Angle < Right Angle < Obtuse Angle < Straight Angle < Reflex Angle".

Complementary Angles
Supplementary Angles
Adjacent Angles
Vertically Opposite Angles
Linear Pair

1. Complementary Angles

Two angles whose sum is 90° (that is, one right angle) are called complementary anglesand one is called the complement of the other.

Here, `/_`AOB = 40° and `/_`BOC = 50°

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» Therefore, `/_`AOB + `/_`BOC = 90°
» Here, `/_`AOB and `/_`BOC are called complementary angles.
» `/_`AOB is complement of `/_`BOC and `/_`BOC is complement of `/_`AOB.

For Example:

(i) Angles of measure 60° and 30° are complementary angles because 60° + 30° = 90°
Thus, the complementary angle of 60° is the angle measure 30°. The complementary angle angle of 30° is the angle of measure 60°.

(ii) Complement of 30° is → 90° - 30° = 60°
(iii) Complement of 45° is → 90° - 45° = 45°
(iv) Complement of 55° is → 90° - 55° = 35°
(v) Complement of 75° is → 90° - 75° = 15°

» Working rule:
To find the complementary angle of a given angle subtract the measure of an angle from 90°.
» So, the complementary angle = 90° - the given angle.


2. Supplementary Angles

» Two angles whose sum is 180° (that is, one straight angle) are called supplementary angles and one is called the supplement of the other.

Here, `/_`PQR = 50° and `/_`RQS = 130°

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» `/_`PQR + `/_`RQS = 180° Hence, `/_`PQR and `/_`RQS are called supplementary angles and `/_`PQR is
» supplement of `/_`RQS and `/_`RQS is supplement of `/_`PQR.

For Example:

(i) Angles of measure 100° and 80° are supplementary angles because 100° + 80° = 180°.

Thus the supplementary angle of 80° is the angle of measure 100°.

(ii) Supplement of - 55° is 180° - 55° = 125°
(iii) Supplement of 95° is 180° - 95° = 85°
(iv) Supplement of 135° is 180° - 135° = 45°
(v) Supplement of 150° is 180° - 150° = 30°


Working rule: To find the supplementary angle of a given angle, subtract the measure of angle from 180°.

» So, the supplementary angle = 180° - the given angle.


3. Adjacent Angles

» Two non - overlapping angles are said to be adjacent angles if they have:

» (a) a common vertex
» (b) a common arm
» (c) other two arms lying on opposite side of this common arm, so that their interiors do not overlap.

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» In the above given figure, `/_`AOB and `/_`BOC are non - overlapping, have OB as the common arm and O as the common vertex.
» The other arms OC and OA of the angles `/_`BOC and `/_`AOB are an opposite sides, of the common arm OB.
» Hence, the arm `/_`AOB and `/_`BOC form a pair of adjacent angles.


4. Vertically Opposite Angles:

» Two angles formed by two intersecting lines having no common arm are called vertically opposite angles.

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» In the above given figure, two lines AB↔- and CD↔-- intersect each other at a point O.
» They form four angles `/_`AOC, `/_`COB, `/_`BOD and `/_`AOD in which `/_`AOC and `/_`BOD are vertically opposite angles. `/_`COB and `/_`AOD are vertically opposite angle.
» `/_`AOC and `/_`COB, `/_`COB and `/_`BOD, `/_`BOD and `/_`DOA, `/_`DOA and `/_`AOC are pairs of adjacent angles.
» Similarly we can say that, `/_`1 and `/_`2 form a pair of vertically opposite angles while `/_`3 and `/_`4 form another pair of vertically opposite angles.
» When two lines intersect, then vertically opposite angles are always equal.

`/_`1 = `/_`2
`/_`3 = `/_`4


5. Linear Pair:

Two adjacent angles are said to form a linear pair if their sum is 180°.

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» These are the pairs of angles in geometry.


Interior and Exterior of an Angle

» The shaded portion between the arms BA and BC of the angle ABC can be extended indefinitely.

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» This portion is called the interior of the angle. X is a point in the interior of the angle. The point Y lies in the exterior of the angle. The point Z lies on the angle.

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» In the above figure, here `/_`1 is called the interior angle because it lies inside the two arms. `/_`2 is called the exterior angle.
» Whenever two rays meet, two angle are formed - one an interior angle and other an exterior angle.
» The size of an angle is measured by the amount of turn or rotation of two arms and not by how long the arms appear to be.

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Here `/_`a is greater than `/_`b, `/_`b is greater than `/_`c.


Bisecting an Angle

» Bisecting an angle means dividing it into two equal angles. The ray which bisects an angle is known as its bisector.

(i) By Paper Folding:

» `/_`LMN is the given angles. Fold the paper so that LM falls along MN and press the paper in this position so as to get an impression.

Unfold the paper. Draw the line along the impression.

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» This line bisects `/_`LMN.

(ii) Using a Protractor:

» `/_`XYZ is to be bisected. Measure `/_` XYZ with the help of protractor. Let it be 72°.
We know half of 72° is 36°.
Make `/_`XYW = 36°, so that YW falls within `/_`XYZ.

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» Now `/_`XYZ is divided into two equal angles by the line YW.

(iii) Using a Compass:

» `/_`ABC is to be bisected. Place the metal end of the compass at B and draw an arc to cut BC at D and AB at E.
Place the metal end at E and D with any convenient radius, draw two arcs (same radius) cutting each other at F.
Join FB.

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» FB is the bisector of `/_`ABC.

Lines of Symmetry

» Learn about lines of symmetry in different geometrical shapes.
» It is not necessary that all the figures possess a line or lines of symmetry in different figures.
» Figures may have:
» No line of symmetry
» 1, 2, 3, 4 ....... lines of symmetry
» Infinite lines of symmetry

Let us consider a list of examples and find out lines of symmetry in different figures:


1. Line segment:

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In the figure there is one line of symmetry. The figure is symmetric along the perpendicular bisector l.


2. An angle:

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» In the figure there is one line of symmetry. The figure is symmetric along the angle bisector OC.


3. An isosceles triangle:

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» In the figure there is one line of symmetry. The figure is symmetric along the bisector of the vertical angle. The median XL.


4. Semi-circle:

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» In the figure there is one line of symmetry. The figure is symmetric along the perpendicular bisector l. of the diameter XY.


5. Kite:

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» In the figure there is one line of symmetry. The figure is symmetric along the diagonal QS.


6. Isosceles trapezium:

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» In the figure there is one line of symmetry. The figure is symmetric along the line l joining the midpoints of two parallel sides AB and DC.


7. Rectangle:

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» In the figure there are two lines of symmetry. The figure is symmetric along the lines l and m joining the midpoints of opposite sides.


8. Rhombus:

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» In the figure there are two lines of symmetry. The figure is symmetric along the diagonals AC and BD of the figure.


9. Equilateral triangle:

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» In the figure there are three lines of symmetry. The figure is symmetric along the 3 medians PU, QT and RS.


10. Square:

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» In the figure there are four lines of symmetry. The figure is symmetric along the 2diagonals and 2 midpoints of opposite sides.


11. Circle:

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» In the figure there are infinite lines of symmetry. The figure is symmetric along all the diameters.

Note:

» Each regular polygon (equilateral triangle, square, rhombus, regular pentagon, regular hexagon etc.) are symmetry.
» The number of lines of symmetry in a regular polygon is equal to the number of sides a regular polygon has.
» Some figures like scalene triangle and parallelogram have no lines of symmetry.


Lines of symmetry in letters of the English alphabet:

» Letters having one line of symmetry:
» A B C D E K M T U V W Y have one line of symmetry.
» A M T U V W Y have vertical line of symmetry.
» B C D E K have horizontal line of symmetry.

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» Letter having both horizontal and vertical lines of symmetry:
» H I X have two lines of symmetry.

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Letter having no lines of symmetry:

» F G J L N P Q R S Z have neither horizontal nor vertical lines of symmetry.

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Letters having infinite lines of symmetry:

  • has infinite lines of symmetry. Infinite number of lines passes through the point symmetry about the center O with all possible diameters.
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Construction of Angles by using Compass

In construction of angles by using compass we will learn how to construct different angles with the help of ruler and compass.

1. Construction of an Angle of 60° by using Compass

Step of Construction:

(i) Draw a ray OA.
(ii) With O as centre and any suitable radius draw an arc above OA cutting it at a point B.
(iii) With B as centre and the same radius as before, draw another arc to cut the previous arc at C.
(iv) Join OC and produce it to D.

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» Then `/_`AOD = 60°.


2. Construction of an Angle of 120° by using Compass

Step of Construction:

(i) Draw a ray OA.
(ii) With O as centre and any suitable radius draw an arc cutting OA at B.
(iii) With B as centre and the same radius cut the arc at C, then with C as centre and same radius cut the arc at D. Join OD and produce it to E.

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» Then, `/_`AOE = 120°.


3. Construction of an Angle of 30° by using Compass

Step of Construction:

(i) Construction an angle `/_`AOD = 60° as shown.
(ii) Draw the bisector OE of `/_`AOD.

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» Then, `/_`AOD = 30°.


4. Construction of an Angle of 90° by using Compass

Step of Construction:

(i) Take any ray OA. (ii) With O as centre and any convenient radius, draw an arc cutting OA at B.
(iii) With B as centre and the same radius, draw an cutting the first arc at C.
(iv) With C as centre and the same radius, cut off an arc cutting again the first arc at D.
(v) With C and D as centre and radius of more than half of CD, draw two arcs cutting each other at E, join OE.

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» Then, `/_`EOA = 90°.


5. Construction of an Angle of 75° by using Compass

Step of Construction:

(i) Take a ray OA.
(ii) With O as centre and any convenient radius, draw an arc cutting OA at C.
(iii) With C as centre and the same radius, draw an cutting the first arc at M.
(iv) With M as centre and the same radius, cut off an arc cutting again the first arc at L.
(v) With L and M as centre and radius of more than half of LM, draw two arcs cutting each other at B, join OB which is making 90°.
(vi) Now with N and M as centres again draw two arcs cutting each other at P.
(vii) Join OP.

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» Then, `/_`POA = 75°.


6. Construction of an Angle of 105° by using Compass

Step of Construction:

(i) After making 90° angle take L and N as centre and draw two arcs cutting each other at S.
(ii) Join SO.

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» Then, `/_`SOA = 105°.


7. Construction of an Angle of 135° by using Compass

Step of Construction:

(i) Construct `/_`AOD = 90°
(ii) Produce `/_`AO to B.
(iii) Draw OE to bisect `/_`DOB.

`/_`DOE = 45°
`/_`EOA = 45° + 90° = 135°

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» Then, `/_`EOA = 135°.


8. Construction of an Angle of 150° by using Compass

Step of Construction:

(i) Construct `/_`AOC = 120°
(ii) Produce `/_`AO to B.
(iii) Draw OD to bisect `/_`COB.

Now `/_`COD = 30°
Therefore, `/_`AOD = 120° + 30° = 150°

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» Then, `/_`AOD = 150°.


Measuring an Angle by a Protractor

In measuring an angle by a protractor, first we need to know what a protractor is.

Protractor

» It is an instrument for measuring or constructing an angle of a given measure. It is a circular or semicircular piece of metal or plastic.

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It is a circular protractor which is marked in degrees from 0° to 360° from left as shown in adjoining figure.

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It is a semicircular protractor which is marked in degrees from 0° to 180° from left as shown in adjoining figure.


For Example:

1. Use your protractor to draw 60°.

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» The centre O of the piece is also the midpoint of its base line. In order to measure `/_`AOB, place the protractor in such a way that its centre is exactly on the vertex O of the angle, the base line lies along the arm OA.
» We need to read the mark through which the arm OB passes, starting from O on the side A, as we observe in the above figure.

Thus we find `/_` AOB = 60°.


2. Use your protractor to draw 90°.

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» The centre O of the piece is also the midpoint of its base line. In order to measure `/_`AOB, place the protractor in such a way that its centre is exactly on the vertex O of the angle, the base line lies along the arm OA.
» We need to read the mark through which the arm OB passes, starting from O on the side A, as we observe in the above figure.

Thus we find `/_` AOB = 90°.

Parallel Lines

Properties of Parallel Lines

What are Parallel Lines?

» Two lines in a plane are said to be parallel if they do not intersect, when extended infinitely in both the direction. Also, the distance between the two lines is the same throughout.

Parallel Lines:

» The symbol for denoting parallel lines is ∥. If lines l and m are parallel to each other, we can write it as l//m and which is read as 'l is parallel to m'.


Properties Of Angles Associated with Parallel Lines
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If two parallel lines are cut by a transversal, then

  • the pair of corresponding angles is equal ( `/_`2 = `/_`6); ( `/_`3 = `/_`7); ( `/_`1 = `/_`5); ( `/_`4 = `/_`8).
  • the pair of interior alternate angles is equal ( `/_`4 = `/_`6); ( `/_`3 = `/_`5).
  • the pair of exterior alternate angles is equal ( `/_`1 = `/_`7); ( `/_`2 = `/_`8).
  • interior angles on the same side of transversal are supplementary, i.e., `/_`3 + `/_`6 = 180° and `/_`4 + `/_`5 = 180°.

F'

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(i) Interior and exterior alternate angles are equal.
i.e. `/_`3 = `/_`6 and `/_`4 = `/_`5 [Interior alternate angles]
`/_`1 = `/_`8 and `/_`2 = `/_`7 [Exterior alternate angles]

(ii) Corresponding angles are equal.
i.e. `/_`1 = `/_`5; `/_`2 = `/_`6; `/_`3 = `/_`7 and `/_`4 = `/_`8

(iii) Co-interior or allied angles are supplementary.
i.e. `/_`3 + `/_`5 = 180° and `/_`4 + `/_`6 = 180°


Conditions of Parallelism:

    If two straight lines are cut by a transversal, and if

  • the pair of corresponding angles is equal, then the two straight lines are parallel to each other.
  • the pair of alternate angles is equal, then the two straight lines are parallel to each other.
  • the pair of interior angles on the same side of transversal is supplementary, then the two straight lines are parallel.
  • Therefore, in order to prove that the given lines are parallel; show either alternate angles are equal or, corresponding angles are equal or, the co-interior angles are supplementary.


Parallel Rays:

  • Two rays are parallel if the corresponding lines determined by them are parallel. In other words, two rays in the same plane are parallel if they do not intersect each other even if extended indefinitely beyond their initial points.
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    Parallel Rays

    Therefore, ray AB ∥ray MN


Parallel Segments:

» Two segments are parallel if the corresponding lines determined by them are parallel.
» In other word, two segments which are in the same plane and do not intersect each other even if extended indefinitely in both directions are said to be parallel.

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Parallel Segments

Therefore, segment AB ∥segment MN

» One segment and one ray are parallel if the corresponding lines determined by them are parallel.

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Parallel Segments

Therefore, segment AB ∥ ray PQ. The opposite edge of a ruler is an example of parallel line segments.


Related Angles

» Related angles are the pairs of angles and specific names are given to the pairs of angles which we come across. These are called related angles as they are related with some condition.

Axiom : If a transversal intersects two parallel lines, then each pair of corresponding angles is equal. Axiom 6.3 is also referred to as the corresponding angles axiom. Now, let us discuss the converse of this axiom which is as follows: If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel. Does this statement hold true? It can be verified as follows: Draw a line AD and mark points B and C on it. At B and C, construct `/_`ABQ and Does this statement hold true? It can be verified as follows: Draw a line AD and mark points B and C on it. At B and C, construct `/_`ABQ and `/_`BCS equal to each other as shown in Fig.

BCS equal to each other as shown in Fig.

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Parallel Segments

Produce QB and SC on the other side of AD to form two lines PQ and RS.
You may observe that the two lines do not intersect each other.
You may also draw common perpendiculars to the two lines PQ and RS at different points and measure their lengths.
You will find it the same everywhere. So, you may conclude that the lines are parallel.
Therefore, the converse of corresponding angles axiom is also true. So, we have the following axiom:


Axiom: If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.
Can we use corresponding angles axiom to find out the relation between the alternate interior angles when a transversal intersects two parallel lines? In Fig. transveral PS intersects parallel lines AB and CD at points Q and R respectively.
Is `/_`BQR = `/_`QRC and `/_`AQR = `/_`QRD?
You know that `/_`PQA = `/_`QRC (1)
(Corresponding angles axiom)

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Parallel Segments

Is `/_`PQA = `/_`BQR? Yes! (Why ?) (2)
So, from (1) and (2), you may conclude that
`/_`BQR = `/_`QRC.
Similarly, `/_`AQR = `/_`QRD.
This result can be stated as a theorem given below:


Theorem 1 : If two lines intersect each other, then the vertically opposite angles are equal.
Proof :
In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. They lead to two pairs of vertically opposite angles, namely,

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Vertically opposite angles

(i) `/_` AOC and `/_` BOD (ii)`/_`AOD and `/_`BOC.
We need to prove that
`/_`AOC = `/_`BOD and `/_` AOD =`/_`BOC.
Now, ray OA stands on line CD.
Therefore, `/_`AOC + `/_`AOD = 180° (Linear pair axiom) .......... (1)
Can we write `/_`AOD + `/_`BOD = 180°? Yes! (Why?)..........(2)
From (1) and (2), we can write
`/_`AOC + `/_`AOD = `/_`AOD + `/_`BOD
This implies that `/_`AOC = `/_`BOD (Refer Section 5.2, Axiom 3)
Similarly, it can be proved that `/_`AOD = `/_`BOC
Now, let us do some examples based on Linear Pair Axiom and Theorem 1.


Theorem 2 : If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.

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Parallel Segments
Now, using the converse of the corresponding angles axiom, can we show the two lines parallel if a pair of alternate interior angles is equal? In Fig, the transversal PS intersects lines AB and CD at points Q and R respectively such that `/_`BQR = `/_`QRC.
Is AB ∥ CD?
`/_`BQR = `/_`PQA (Why?) (1)
But,`/_`BQR = `/_`QRC (Given) (2)
So, from (1) and (2), you may conclude that
`/_`PQA = `/_`QRC But they are corresponding angles.
So, AB ∥ CD (Converse of corresponding angles axiom)
This result can be stated as a theorem given below:


Theorem 3 : If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
In a similar way, you can obtain the following two theorems related to interior angles on the same side of the transversal.


Theorem 4 : If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.


Theorem 5 : If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.
You may recall that you have verified all the above axioms and theorems in earlier classes through activities. You may repeat those activities here also.


Lines Parallel to the Same Line

If two lines are parallel to the same line, will they be parallel to each other? Let us check it. See Fig.in which line m ∥line l and line n ∥ line l.

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Let us draw a line t transversal for the lines, l, m and n. It isgiven thatline m ∥ line l and line n ∥ line l.
Therefore, `/_`1 = `/_`2 and `/_`1 = `/_`3 (Corresponding angles axiom)
So,`/_`2 = `/_`3 (Why?)
But `/_`2 and `/_`3 are corresponding angles and they are equal.
Therefore, you can say thatLine m ∥ Line n (Converse of corresponding angles axiom)
This result can be stated in the form of the following theorem:


Theorem 6 : Lines which are parallel to the same line are parallel to each other.
Note : The property above can be extended to more than two lines also. Now, let us solve some examples related to parallel lines.

Angle Sum Property of a Triangle

In the earlier classes, you have studied through activities that the sum of all the angles of a triangle is 180°. We can prove this statement using the axioms and theorems related to parallel lines.

Theorem 7 :
The sum of the angles of a triangle is 180°.
Proof :
Let us see what is given in the statement above, that is, the hypothesis and what we need to prove. We are given a triangle PQR and `/_`1, `/_`2and `/_`3 are the angles of `Delta`PQR (see Fig.).
We need to prove that `/_`1 +`/_`2 + `/_`3 = 180°. Let us draw a line XPY parallel to QR through the opposite vertex P, as shown in Fig., so that we can use the properties related to parallel lines.
Now, XPY is a line.
Therefore,`/_`4 + `/_`1 + `/_`5 = 180°........(1)
But XPY || QR and PQ, PR are transversals.
So,`/_`4 = `/_`2and `/_`5 = `/_`3(Pairs of alternate angles)
Substituting `/_`4 and `/_`5 in (1),
we get`/_`2 + `/_`1 + `/_`3 = 180°
That is,`/_`1 + `/_`2 + `/_`3 = 180°
„Recall that you have studied about the formation of an exterior angle of a triangle in the earlier classes (see Fig. 6.36). Side QR is produced to point S, `/_`PRS is called an exterior angle of `Delta`PQR.
Is `/_`3 + `/_`4 = 180°? (Why?).......... (1)
Also, see that `/_`1 + `/_`2 + `/_`3 = 180° (Why?).......... (2)
From (1) and (2), you can see that `/_`4 = `/_`1 + `/_`2.
This result can be stated in the form of a theorem as given below:
Theorem 8 : If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
It is obvious from the above theorem that an exterior angle of a triangle is greater than either of its interior apposite angles.
Now, let us solve some examples based on the above theorems

Complementary and Supplementary Angles

Complementary angles:

»     When the sum of the measures of two angles is 90°, such angles are called complementary angles.

For example:

»     An angle of 30° and another angle of 60° are complementary angles of each other.
Also, complement of 30° is 90° - 30° = 60°.
And complement of 60° is 90° - 60° = 30°

`/_`AOB + `/_`POQ = 90°


Supplementary angles:

»     When the sum of the measures of two angles is 180°, such angles are called supplementary angles.

For example:

»     An angle of 120° and another angle of 60° are supplementary angles of each other. Also, supplement of 120° is 180° - 120° = 60°. And supplement of 60° is 180° - 60° = 120°

lines59

`/_`AOB + `/_`POQ = 180°


Adjacent angles:

»     Two angles in a plane are said to be adjacent if they have a common arm, a common vertex and the non-common arms lie on the opposite side of the common arm.

lines60

»     In the given figure, `/_`AOC and `/_`BOC are adjacent angles as OC is the common arm, O is the common vertex, and OA, OB are on the opposite side of OC.


Linear pair:

»     Two adjacent angles form a linear pair of angles if their non-common arms are two opposite rays, i.e., the sum of two adjacent angles is 180°.
Here, `/_`AOB + `/_`AOC
= 180°

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Vertically opposite angles:

»     When two lines intersect, then the angles having their arms in the opposite direction are called vertically opposite angles.

»     The pair of vertically opposite angles is equal.
Here the pairs of vertically opposite angles are `/_`AOD and `/_`BOC, `/_`AOC and `/_`BOD.

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1. If a ray stands on a line, then the sum of adjacent angles formed is 180°.

Given: A ray RT standing on (PQ) such that `/_`PRT and `/_`QRT are formed.

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Construction: Draw RS ⊥ PQ.
Proof: Now `/_`PRT = `/_`PRS + `/_`SRT ......... (1)
Also `/_`QRT = `/_`QRS - `/_`SRT ......... (2)

Adding (1) and (2),

`/_`PRT + `/_`QRT = `/_`PRS + `/_`SRT + `/_`QRS - `/_`SRT
= `/_`PRS + `/_`QRS
= 90° + 90°
= 180°


2. The sum of all the angles around a point is equal to 360°.

Given: A point O and rays OP, OQ, OR, OS, OT which make angles around O.

`/_`PRT + `/_`QRT = `/_`PRS + `/_`SRT + `/_`QRS - `/_`SRT
             = `/_`PRS + `/_`QRS
             = 90° + 90°
             = 180°

Construction: Draw OX opposite to ray OP
Proof: Since, OQ stands on XP therefore
`/_`POQ + `/_`QOX = 180°
`/_`POQ + ( `/_`QOR + `/_`ROX) = 180°
`/_`POQ + `/_`QOR + `/_`ROX = 180°......... (i)

Again OS stands on XP, therefore
`/_`XOS + `/_`SOP = 180°
`/_`XOS + ( `/_`SOT + `/_`TOP) = 180°
`/_`XOS + `/_`SOT + `/_`TOP = 180°......... (ii)

Adding (i) and (ii),

`/_`POQ + `/_`QOR + `/_`ROX + `/_`XOS + `/_`SOT + `/_`TOP
= 180° + 180°
= 360°


3. If two lines intersect, then vertically opposite angles are equal.

Given: PQ and RS intersect at point O.

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Proof: OR stands on PQ.
Therefore, `/_`POR + `/_`ROQ = 180°......... (i)
PO stands on RS
`/_`POR + `/_`POS = 180°......... (ii)

From (i) and (ii),

`/_`POR + `/_`ROQ = `/_`POR + `/_`POS
`/_`ROQ + `/_`POS
Similarly, `/_`POR = `/_`QOS can be proved.


Complementary and Supplementary Angles

Before we solve the worked-out problems on complementary and supplementary angles we will recall the definition of complementary angles and supplementary angles.

Complementary Angles:
  • Two angles are called complementary angles, if their sum is one right angle i.e. 90°. Each angle is called the complement of the other.
  • Example, 20° and 70° are complementary angles, because 20° + 70° = 90°.
  • Clearly, 20° is the complement of 70° and 70° is the complement of 20°.
  • Thus, the complement of angle 53° = 90° - 53° = 37°.

Supplementary Angles:
  • Two angles are called supplementary angles, if their sum is two right angles i.e. 180°. Each angle is called the supplement of the other.
  • Example, 30° and 150° are supplementary angles, because 30° + 150° = 180°
  • .
  • Clearly, 30° is the supplement of 150° and 150° is the supplement of 30°.
  • Thus, the supplement of angle 105° = 180° - 105° = 75°.

Solved problems on complementary and supplementary angles

1. Find the complement of the angle 2/3 of 90°.

Solution:

Convert 2/3 of 90°
2/3 × 90° = 60°
Complement of 60° = 90° - 60° = 30°
Therefore, complement of the angle 2/3 of 90° = 30°


2. Find the supplement of the angle 4/5 of 90°.

Solution:

Convert 4/5 of 90°
4/5 × 90° = 72°
Supplement of 72° = 180° - 72° = 108°
Therefore, supplement of the angle 4/5 of 90° = 108°


3. The measure of two complementary angles are (2x - 7)° and (x + 4)°. Find the value of x.

Solution:

According to the problem, (2x - 7)° and (x + 4)°, are complementary angles’ so we get;
(2x - 7)° + (x + 4)° = 90°
or, 2x - 7° + x + 4° = 90°
or, 2x + x - 7° + 4° = 90°
or, 3x - 3° = 90°
or, 3x - 3° + 3° = 90° + 3°
or, 3x = 93°
or, x = 93°/3°
or, x = 31°
Therefore, the value of x = 31°.


4. The measure of two supplementary angles are (3x + 15)° and (2x + 5)°. Find the value of x.

Solution:

According to the problem, (3x + 15)° and (2x + 5)°, are complementary angles’ so we get;
(3x + 15)° + (2x + 5)° = 180°
or, 3x + 15° + 2x + 5° = 180°
or, 3x + 2x + 15° + 5° = 180°
or, 5x + 20° = 180°
or, 5x + 20° - 20° = 180° - 20°
or, 5x = 160°
or, x = 160°/5°
or, x = 32°
Therefore, the value of x = 32°.


5. The difference between the two complementary angles is 180°. Find the measure of the angle.

Solution:

Let one angle be of measure x°.
Then complement of x° = (90 - x)
Difference = 18°

Therefore, (90° - x) – x = 18°
or, 90° - 2x = 18°
or, 90° - 90° - 2x = 18° - 90°
or, -2x = -72°
or, x = 72°/2°
or, x = 36°

Also, 90° - x
= 90° - 36°
= 54°.

Therefore, the two angles are 36°, 54°.


6. POQ is a straight line and OS stands on PQ. Find the value of x and the measure of `/_` POS, `/_` SOR and `/_` ROQ.

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Solution:

POQ is a straight line.
Therefore, `/_`POS + `/_`SOR + `/_`ROQ = 180°
or, (5x + 4°) + (x - 2°) + (3x + 7°) = 180°
or, 5x + 4° + x - 2° + 3x + 7° = 180°
or, 5x + x + 3x + 4° - 2° + 7° = 180°
or, 9x + 9° = 180°
or, 9x + 9° - 9° = 180° - 9°
or, 9x = 171°
or, x = 171/9
or, x = 19°

Put the value of x = 19°
Therefore, x - 2
= 19 - 2
= 17°

Again, 3x + 7
= 3 × 19° + 7°
= 570 + 7°
= 64°

And again, 5x + 4
= 5 × 19° + 4°
= 95° + 4°
= 99°

Therefore, the measure of the three angles is 17°, 64°, 99°.


These are the above solved examples on complementary and supplementary angles explained step-by-step with detailed explanation.

Parallel and Transversal Lines

    When the transversal intersects two parallel lines:

  • Pairs of corresponding angles are equal.
  • Pairs of alternate angles are equal
  • Interior angles on the same side of transversal are supplementary.

Worked-out problems for solving parallel and transversal lines

1. In adjoining figure l ∥ m is cut by the transversal t. If `/_`1 = 70, find the measure of `/_`3, `/_`5, `/_`6.

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Solution:

We have `/_`1 = 70°
`/_`1 = `/_`3 (Vertically opposite angles)
Therefore, `/_`3 = 70°

Now, `/_`1 = `/_`5 (Corresponding angles)
Therefore, `/_`5 = 70°

Also, `/_`3 + `/_`6 = 180° (Co-interior angles)
70° + `/_`6 = 180°
Therefore, `/_`6 = 180° - 70° = 110°


2. In the given figure AB ∥ CD, `/_`BEO = 125°, `/_`CFO = 40°. Find the measure of `/_`EOF.

Solution:

lines74

»     Draw a line XY parallel to AB and CD passing through O such that AB ∥ XY and CD ∥ XY

»     `/_`BEO + `/_`YOE = 180° (Co-interior angles)
Therefore, 125° + `/_`YOE = 180°

»     Therefore, `/_`YOE = 180° - 125° = 55°
»     Also, `/_`CFO = `/_`YOF (Alternate angles)

»     Given `/_`CFO = 40°
Therefore, `/_`YOF = 40°

»     Then `/_`EOF = `/_`EOY + `/_`FOY
= 55° + 40° = 95°


3. In the given figure AB ∥ CD ∥ EF and AE ⊥ AB. Also, `/_`BAE = 90°. Find the values of `/_`x, `/_`y and `/_`z.

Solution:

lines75

»     y + 45° = 1800
Therefore, `/_`y = 180° - 45° (Co-interior angles)
= 135°

»     `/_`y = `/_`x (Corresponding angles)
Therefore, `/_`x = 135°

»     Also, 90° + `/_`z + 45° = 180°
Therefore, 135° + `/_`z = 180°

»     Therefore, `/_`z = 180° - 135° = 45°


4. In the given figure, AB ∥ ED, ED ∥ FG, EF ∥ CD Also, `/_`1 = 60°, `/_`3 = 55°, then find `/_`2, `/_`4, `/_`5.

Solution:

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»     Since, EF ∥ CD cut by transversal ED
Therefore, `/_`3 = `/_`5 we know, `/_`3 = 55°
Therefore, `/_`5 = 55°

»     Also, ED ∥ XY cut by transversal CD
Therefore, `/_`5 = `/_`x we know `/_`5 = 55°

»     Therefore, `/_`x = 55°

»     Also, `/_`x + `/_`1 + `/_`y = 180°
55° + 60° + `/_`y = 180°
115° + `/_`y = 180°
`/_`y = 180° - 115°
Therefore, `/_`y = 65°

»     Now, `/_`y + `/_`2 = 180°(Co-interior angles)

lines77

`/_`y = 65°
`/_`2 = 180° - `/_`Y
»     65° + `/_`2 = 180°
`/_`2 = 180° - 65°
`/_`2 = 115°

»     Since, ED ∥ FG cut by transversal EF

Therefore, `/_`3 + `/_`4 = 180°
55° + `/_`4 = 180°
Therefore, `/_`4 = 180° - 55° = 125°


5. In the given figure PQ ∥ XY. Also, y : z = 4 : 5 find.

lines78

Solution:

»     Let the common ratio be a
Then y = 4a and z = 5a
Also, `/_`z = `/_`m (Alternate interior angles)

»     Since, z = 5a
Therefore, `/_`m = 5a [RS ∥ XY cut by transversal t]

»     Now, `/_`m = `/_`x (Corresponding angles)
Since, `/_`m = 5a
Therefore, `/_`x = 5a [PQ ∥ RS cut by transversal t]

»     `/_`x + `/_`y = 180° (Co-interior angles)

»     5a + 4a = 1800
9a = 180°
a = 180/9
a = 20

»     Since, y = 4a
Therefore, y = 4 × 20
y = 80°

»     z = 5a
Therefore, z = 5 × 20
z = 100°

»     x = 5a
Therefore, x = 5 × 20
x = 100°

Therefore, `/_`x = 100°, `/_`y = 80°, `/_`z = 100°

Fundamental Geometrical Concepts

  • The fundamental geometrical concepts depend on three basic concepts — point, line and plane. The terms cannot be precisely defined. However, the meanings of these terms are explained through examples.

Point:

    lines79a
  • It is the mark of position and has an exact location.
  • It has no length, breadth or thickness.
  • It is denoted by a dot made by the tip of a sharp pencil.
  • It is denoted by capital letter.
  • In the given figure P, Q, R represents different points.

Line:

    lines79
  • It is a straight path which can be extended indefinitely in both the directions.
  • It is shown by two arrowheads in opposite directions.
  • It does not have any fixed length.
  • It has no endpoints.
  • It is denoted as AB or BA and is read as line AB or line in the BA.
  • It can never be measured.
  • Infinite number of points lie on the line.
  • Sometimes it is also denoted by small letters of the English alphabet.

Ray:

    lines80
  • It is a straight path which can be extended indefinitely in one direction only and the other end is fixed.
  • It has no fixed length.
  • It has one endpoint called the initial point.
  • It cannot be measured.
  • It is denoted as OA and is read as ray OA.
  • A number of rays can be drawn from an initial point O.
  • Ray OA and ray OB are different because they are extended in different directions.
  • Infinite points lie on the ray.

Line Segment:

    lines81
  • It is a straight path which has a definite length.
  • It has two endpoints.
  • It is a part of the line.
  • It is denoted as AB or BA.
  • It is read as line segment AB or line segment BA.
  • The distance between A and B is called the length of AB.
  • Infinite number of points lies on a line segment.
  • Two line segments are said to be equal if they have the same length.

Plane:

    lines82
  • A smooth, flat surface gives us an idea of a plane. The surface of the table, wall, blackboard, etc., is smooth and flat. It extends endlessly in all the directions. It has no length, breadth or thickness.
  • Here, we have shown a portion of a particular plane. We can draw certain figures like square, rectangle, triangle, and circle on the plane. Hence, these figures are also called plane figures.

Incidence Properties of Lines in a Plane:

  • An infinite number of many lines can be drawn to pass through a given point in a plane. Through a given point in a plane, infinitely many lines can be drawn to pass through.
  • lines83
  • Two distinct points in a plane determine a unique line.
    One and only one line can be drawn to pass through two given points, i.e., two distinct points in a plane. This line lies wholly in the plane.
  • lines84
  • Infinite number of points lie on the line in a plane.
  • lines85
  • Two lines in a plane either intersect at a point or they are parallel to each other.
  • lines86

Collinear Points:

    lines87

    Two or more points which lie on the same line in a plane are called collinear points.

  • The line is called the line of colinearity.
  • Two points are always collinear.
  • In the adjoining figure......
    Points A, B, C are collinear lying on line
    Points X, Y, Z are not collinear because all the three points do not lie on a line.
    Hence, they are called non-collinear points.
  • lines88

    Similarly, here points M, N, O, P, Q are collinear points and A, B, C are non-collinear points.

    lines89

    Note:
    Two points are always collinear.


Concurrent Lines:

lines90

Three or more lines which pass through the same point are called concurrent lines and this common point is called the point of concurrence. In the adjoining figure, lines p, q, r, s, t, u intersect at point O and are called concurrent lines.


Two lines in a Plane:

Intersecting Lines: Two lines in a plane which cut each other at common point are called intersecting lines and the point is called the point of intersection. In the adjoining figure, lines l and m intersect at point O.

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Parallel Lines: Two lines in a plane which do not intersect at any point, i.e., they do not have any point in common are called parallel lines. The distance between the two parallel lines remains the same throughout.

lines92

These are the fundamental geometrical concepts explained above using figures.

  1. In this chapter, you have studied the following points:
  2. If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and vice-versa. This property is called as the Linear pair axiom.
    If two lines intersect each other, then the vertically opposite angles are equal.
  3. If a transversal intersects two parallel lines, then
    (i) each pair of corresponding angles is equal,
    (ii) each pair of alternate interior angles is equal,
    (iii) each pair of interior angles on the same side of the transversal is supplementary.
  4. If a transversal intersects two lines such that, either
    (i) any one pair of corresponding angles is equal, or
    (ii) any one pair of alternate interior angles is equal, or
    (iii) any one pair of interior angles on the same side of the transversal is supplementary, then the lines are parallel.
    Lines which are parallel to a given line are parallel to each other.
  5. The sum of the 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.

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