SUB MATHEMATICS CLASS VIII NAME OF THE TOPIC

SUB: MATHEMATICS CLASS : VIII NAME OF THE TOPIC : PARALLEL LINES Work is Worship

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LEARNING OBJECTIVES After this lesson, students will be able to: • identify and describe parallel lines. • practice various methods of proving that lines are parallel. • choose a method of measuring angles to prove lines are parallel. • divide a line segment into equal parts as well as in a given ratio internally. Work is Worship

INTRODUCTION • Do you know what parallel lines are? You will understand this with the following examples. • Every one of you must have seen the pair of railway tracks or a ladder or zebra crossing or opposite edges of your book. • What is one common thing among all these? RAILWAY TRACK ZEBRA CROSSING LADDER BOOK Work is Worship

WHAT DO YOU OBSERVE ? • The two tracks never meet each other. Also the two sides of the ladder never intersect each other. The lines in a zebra crossing never meet each other. Opposite edges of your book also never intersect each other. • Let us take another example which you all are using everyday while solving problems in Mathematics. Yes, that is your commonly used ‘ = ‘ symbol( equal symbol) • The two line segments in this symbol also never intersect each other. • SUCH TYPE OF LINES WHICH DO NOT INTERSECT EACH OTHER , ARE CALLED PARALLEL LINES. • • • Work is Worship

PARALLEL LINES AND NON-PARALLEL LINES • The lines on the same plane which do not intersect each other even if extended in any direction, are called parallel lines. • There is no common point between two parallel lines. • Example: Opposite sides of a rectangle. NON – PARALLEL LINES • Lines which are not parallel, are called non parallel or intersecting lines. • There is a common point between two non-parallel lines. • Example: Adjacent sides of a rectangle. Work is Worship

PARALLEL LINES AND THEIR TRANSVERSAL • When a line intersects two or more parallel lines at distinct points, then that line is called TRANSVERSAL of the lines. • In the figure, line t Intersects line r and s at two distinct points. So, line t is a transversal of line r and s. • Symbol used to represent parallel lines: Work is Worship

ANGLES FORMED BY A TRANSVERSAL WITH TWO PARALLEL LINES • When a transversal intersects two parallel lines, at each point of intersection four angles will be formed. • So , if two parallel lines get intersected by a transversal, eight angles will be formed. • In the given figure, s ∥ t and r is the transversal. Eight angles formed are named as ∠ 1, ∠ 2, ∠ 3, ∠ 4, ∠ 5, ∠ 6, ∠ 7 and ∠ 8. • Let us consider the details in a tabular form for easy reference. Types of Angles Interior Angles ∠ 3, ∠ 4, ∠ 5, ∠ 6 Exterior Angles ∠ 1, ∠ 2, ∠ 7, ∠ 8 Vertically opposite Angles (∠ 1, ∠ 3), (∠ 2, ∠ 4), (∠ 5, ∠ 7), (∠ 6, ∠ 8) Corresponding Angles (∠ 1, ∠ 5), (∠ 2, ∠ 6), (∠ 3, ∠ 7), (∠ 4, ∠ 8) Interior Alternate Angles (∠ 3, ∠ 5), (∠ 4, ∠ 6) Exterior Alternate Angles (∠ 1, ∠ 7), (∠ 2, ∠ 8) Interior Angles on the same side of transversal (∠ 3, ∠ 6), (∠ 4, ∠ 5) Work is Worship

PROPERTIES OF PARALLEL LINES When a transversal intersects two parallel lines ⇛The corresponding angles are equal. (∠ 1= ∠ 5), (∠ 2=∠ 6), (∠ 3= ∠ 7), (∠ 4=∠ 8) ⇛The alternate interior angles are equal. (∠ 3 =∠ 5), (∠ 4 = ∠ 6) ⇛The alternate exterior angles are equal. (∠ 1=∠ 7), (∠ 2=∠ 8) ⇛The pair of interior angles on the same side of the transversal are supplementary. ∠ 3+∠ 6 = 180° and ∠ 4 + ∠ 5 = 180° Work is Worship

LET’S TRY ………. Q. In the given figure , a ∥ b. If ∠ 1 = 55° , find ∠ 2, ∠ 5 and ∠ 8. Answer : ∠ 1+∠ 2 = 180° (linear pair) ∴ ∠ 2 = 180° - ∠ 1 = 180° - 55° = 125° ∠ 1= ∠ 5 (pair of corresponding angles) ∴ ∠ 5 = 55° ∠ 2=∠ 8 (alternate exterior angles) ∴ ∠ 8 = 125° Thus , ∠ 2 = 125° , ∠ 5 = 55° and ∠ 8 = 125°. Work is Worship

SAMPLE QUESTION WITH BREAK UP MARKS Q. In the given figure , a ∥ b. If ∠ 1 = 55° , find ∠ 2, ∠ 5 and ∠ 8. (3 marks) Answer : ∠ 1+∠ 2 = 180° (linear pair) …………… [0. 5] ∴ ∠ 2 = 180° - ∠ 1 = 180° - 55° = 125° ………………[0. 5] ∠ 1= ∠ 5 (pair of corresponding angles) ……. . [0. 5] ∴ ∠ 5 = 55° …………………. [0. 5] ∠ 2=∠ 8 (alternate exterior angles) …………………[0. 5] ∴ ∠ 8 = 125° ………………………. [0. 5] Thus , ∠ 2 = 125° , ∠ 5 = 55° and ∠ 8 = 125°. N. B. If you will not write the reason , as a whole 1 mark will be deducted Work is Worship

CONDITIONS FOR LINES TO BECOME PARALLEL • If a transversal cuts a pair of lines in such a way that – ⇛the angles of any pair of corresponding angles are equal, then the lines are parallel. ⇛the angles of any pair of alternate interior (or exterior) angles are equal, then the lines are parallel. ⇛The angles of any pair of interior angles on the same side of the transversal are supplementary, then the lines are parallel. Work is Worship

LET’S TRY………. Q. In the given figure, ∠ 1= ∠ 2. Show that AD∥ BC. [2 marks] Answer: D C Given that ∠ 1= ∠ 2 But they forms a pair of 1 2 corresponding angles. [1] A B As corresponding angles are equal, lines are parallel. ∴ AD∥BC. [0. 5] Work is Worship

Q. In the given figure, ∠ 1=135°, ∠ 2=45° and ∠ 3=135°, , verify that l ∥m and p∥ q. Give reason. (2 marks) • Answer : l m ∠ 1=135° and ∠ 2=45° 1 4 Then ∠ 4 = 180° - 45° =135° (linear pair) 2 p ∠ 1 = ∠ 4 = 135° …………. (0. 5) 3 q ⇒l ∥ m as p is transversal and one pair of corresponding angles is equal. …………………. (0. 5) Again, ∠ 1+ ∠ 2 = 135°+ 45°= 180° …………. . (0. 5) As one pair of interior angles on the same side of transversal is supplementary , the lines are parallel. ⇒p ∥ q …………………(0. 5) N. B. If you will not write the reason, as a whole 0. 5 mark Work is will Worship

DISTANCE BETWEEN PARALLEL LINES • The distance between parallel lines does not change. It is the same every where. s A P t B s ∥ t ⇒ AB= PQ Q Work is Worship

LINES PARALLEL TO A GIVEN LINE • Lines, parallel to a given lines in a plane, are parallel to each other. • In the given figure, 1 l l∥ m and l∥ n so ∠ 1= ∠ 2 [corresponding angles 2 m as l∥ m ] 3 n and ∠ 1= ∠ 3 [corresponding angles as l∥ n ] ∴ ∠ 1= ∠ 2 = ∠ 3 and these forms a pair of corresponding angles. ∴ l∥m∥n Work is Worship

LINES PERPENDICULAR TO A GIVEN LINE • Lines, perpendicular to a given line in a plane, are parallel to each other. • In the given figure, m n m⊥ l and n⊥ l so ∠ 1= 90° and ∠ 2 = 90° 1 2 l so, ∠ 1= ∠ 2 = 90° and these forms a pair of corresponding angles. ∴ m∥n Work is Worship

DIVISION OF A LINE SEGMENT INTO EQUAL PARTS Q. Draw a line segment AB= 6 cm and divide it into five equal parts. Ans. Step –I : Draw a line segment AB = 6 cm. Step –II : At A , draw a ray AC making an angle with AB Step –III : At B , draw a ray BD parallel to AP on the opposite side of AB Step –IV : Using compass, mark five points on AC at equal distances and five points on BD with the same distances. Step –V : Join the points as shown in the figure. The line segment now divided into five equal parts. • https: //drive. google. com/file/d/1 C 3 HCRKfw. OCzj. FRPSNks. URd. Hj. Hn 5 Tk. JXJ/ view? usp=drivesdk Work is Worship

DIVISION OF A LINE SEGMENT IN A GIVEN RATIO INTERNALLY Q. Draw a line segment AB= 5. 5 cm and divide it internally in the ratio 4: 5. (4 marks) Ans. Step –I : Draw a line segment AB = 5. 5 cm. [0. 5] Step –II : At A , draw a ray AX making an acute angle with AB [0. 5] Step –III : At B , draw a ray BY parallel to AX on the opposite side of AB [1] Step –IV : Using compass, mark four points on AX at equal distances and five points on BY with the same distances. [1] Step –V : Join the last points on AX and BY as shown in the figure. The line segment AB now divided in 4: 5 ratio internally. [1] Work is Worship

ACTIVITY • https: //youtu. be/Wit. F_d. WOm. No Work is Worship

ART INEGRATION • Using parallel lines, we can design beautiful arts. • https: //drive. google. com/file/d/1 C 0 -A 86 KLF 1 F 5 Nd. F 3 Ko. Nm 3 r. EEzv 5 JCNw/view? usp=drivesdk Work is Worship

APPLICATION OF PARALLEL LINES IN TESSELLATION Do you know what is Tessellation ? A tessellation is a simple process of tiling a floor or surface with a definite shape repeated and joined together so that there is no overlapping or gaps. Work is Worship

APPLICATION OF PARALLEL LINES IN TESSELLATION (CONTD. ) • We can make beautiful TESSELLATIONS using parallel lines. Work is Worship

KEY CONCEPT • Parallel lines always remain in the same plane and never intersect each other. • The distance between parallel lines is same every where. • Lines, parallel to a given line in a plane, are parallel to each other. • Lines, perpendicular to a given line in a plane, are parallel to each other. • When a transversal intersects two parallel lines ⋆the corresponding angles are equal ⋆the alternate interior angles are equal ⋆the alternate exterior angles are equal ⋆the pair of interior angles on the same side of the transversal are supplementary Work is Worship

KEY CONCEPT(CONTD…) • If a transversal cuts a pair of lines in such a way that – ⋆ the angles of any pair of corresponding angles are equal, then the, lines are parallel ⋆ the angles of any pair of alternate interior (or exterior) angles are equal, then the lines are parallel. ⋆the angles of any pair of interior angles on the same side of the transversal are supplementary, then the lines are parallel. • A line segment can be divided into equal parts by constructing parallel lines. • A line segment can be divided internally in a given ratio by constructing parallel lines. Work is Worship

MIND MAPPING PROPERTIES OF PARALLEL LINES CONDITION FOR LINES TO BECOME PARALLEL CORRESPONDING ANGLES ARE EQUAL ALTRENATE INTERIOR (OR EXTERIOR) ANGLES ARE EQUAL ANGLES ON THE SAME SIDE OF THE TRANSVERSAL ARE SUPPLIMENTARY IN THE SAME PLANE LINES PARALLEL TO A GIVEN LINE ARE PARALLEL TO EACH OTHER IF A PAIR OF CORRESPONDING ANGLES ARE EQUAL DISTANCE BETWEEN TWO PARALLEL LINES DOES NOT CHANGE IN THE SAME PLANE LINES PERPENDICULR TO A GIVEN LINE ARE PARALLEL TO EACH OTHER IF A PAIR OF ALTERNATE INTERIOR ( OR EXTERIOR) ANGLES ARE EQUAL IF A PAIR OF INTERIOR ANGLES ON THE SAME SIDE OF THE TRANSVERSAL ARE SUPPLIMENTARY Work is Worship

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