{"id":623860,"date":"2023-08-31T11:50:42","date_gmt":"2023-08-31T11:50:42","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=623860"},"modified":"2023-08-31T11:51:33","modified_gmt":"2023-08-31T11:51:33","slug":"kc-sinha-exercise-8-2-mathematics-solution-class-9-chapter-8-lines-and-angles","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-2-mathematics-solution-class-9-chapter-8-lines-and-angles\/","title":{"rendered":"KC Sinha: Exercise 8.2 – Mathematics Solution Class 9 Chapter 8 Lines and Angles"},"content":{"rendered":"\n\n\n\n\n

Exercise 8.2<\/h3>\n\n\n\n

Type 1<\/p>\n\n\n\n

Question 1 <\/h4>\n\n\n\n

Answer the following questions :<\/strong>
(i) If any transversal intersects two parallel lines in such a way that the ratio of interior angles on the same side is 2 : 7 , then f\u200cind the measure of the bigger angle in degrees.<\/strong>
(ii) If two parallel lines are intersected by a transversal, what relation will exist between alternate angles ?<\/strong>
(iii) If each of the two lines is parallel to a third line, what relation exists between them ?<\/strong>
(iv) At what angle, in degree measure, all perpendiculars drawn to a line segment are inclined to one another ?<\/strong>
(v) If P be a point not lying on line ‘l’ then how many lines parallel to ‘l’ through the point P, can be drawn ?<\/strong>
(vi) If a transversal intersects two parallel lines, then write two conditions for these two lines to be parallel.<\/strong>
(vii) If arms of any two angles are parallel, what relation exists between these angles ?<\/strong>
Sol :<\/strong>
(i) 2:7
2x+7x=180\u00b0
9x=180\u00b0
x=\\frac{180^{\\circ}}{9}
x=20\u00b0<\/p>\n\n\n\n

bigger angle-
7x=7\u00d720\u00b0
=140\u00b0<\/p>\n\n\n\n

(ii) will be equal<\/p>\n\n\n\n

(iii) they will be Parallel<\/p>\n\n\n\n

(iv) 0\u00b0 or 180\u00b0<\/p>\n\n\n\n

(v) only one<\/p>\n\n\n\n

(vi) Pair of alternate interior angles are equal; pair of corresponding angles are equal<\/p>\n\n\n\n

(vii) will be parallel<\/p>\n\n\n\n

\n\n\n\n

Question 2<\/h4>\n\n\n\n

Fill up the blanks :<\/strong>
(i) All perpendiculars drawn on a line segment are __<\/strong>
(ii) If a transversal intersects two lines in such a way that a pair of alternate interior angles are equal, then the two lines are __<\/strong>
(iii) A transversal intersects two parallel lines and one pair of alternate angles x and y are formed , then x and y are __<\/strong>
Sol :
(i) Parallel
(ii) Parallel
(iii) Equal<\/p>\n\n\n\n

\n\n\n\n

Question 3<\/h4>\n\n\n\n

P is a point between two parallel lines AB and CD. If \u2220ABP=30\u00b0 , \u2220CDP=45\u00b0 , then find the value of \u2220DPB.<\/strong>
Sol :
Diagram
Construction: Draw a line RS through P parallel to the AB and CD line.<\/p>\n\n\n\n

Given: AB||CD
\u2220ABP=30\u00b0,\u2220CDP=45\u00b0<\/p>\n\n\n\n

To prove: \u2220DPB=?<\/p>\n\n\n\n

PS||AB(from construction)<\/p>\n\n\n\n

\u2220SPB=\u2220ABP(alternate interior angle)<\/p>\n\n\n\n

\u2235RS||AB
AB||CD(given)<\/p>\n\n\n\n

\u2234RS||CD
\u2220SPD=\u2220CDP(alternate interior angles)
\u2220SPD=45\u00b0<\/p>\n\n\n\n

\u2220DPB=\u2220SPB+\u2220SPD
=30\u00b0+45\u00b0
=75\u00b0<\/p>\n\n\n\n

\n\n\n\n

Type 2<\/p>\n\n\n\n

Question 4<\/h4>\n\n\n\n

In the given f\u200cigure, if AB||DE, then find the value of \u2220BCD<\/strong>
<figure to be added>
[Hint: Draw a line PCQ through C parallel to AB and DE ]<\/p>\n\n\n\n

Sol:
Diagram
Given: AB||DE<\/p>\n\n\n\n

\u2220ABC=100\u00b0,\u2220CDE=120\u00b0
\u2220BCD=?<\/p>\n\n\n\n

Construction: We can extend the line AB towards the line CD.<\/p>\n\n\n\n

Now,
\u2220ABC+\u2220CBP=180\u00b0(Linear pair)
100\u00b0+\u2220CBP=180\u00b0
\u2220CBP=80\u00b0<\/p>\n\n\n\n

\u2220CBP+\u2220BCP=\u2220CDE(exterior angle is equal to the sum of opposite interior angles)
80\u00b0+\u2220BCD=120\u00b0 (\u2235\u2220CPT=\u2220PDE or \u2220CPT=\u2220CDE)
\u2220BCD=40\u00b0<\/p>\n\n\n\n

\n\n\n\n

Question 5<\/h4>\n\n\n\n

In the given figure, if \u22201=130\u00b0 and \u22206=60\u00b0 , is AB||CD ?<\/strong>
<figure to be added>
Sol :
Diagram
\u22205+\u22206=180\u00b0(linear pair)
\u22205+60\u00b0=180\u00b0<\/p>\n\n\n\n

\u22205=120\u00b0<\/p>\n\n\n\n

\u22201\u22605<\/p>\n\n\n\n

\u2234AB\u2226CD<\/p>\n\n\n\n

\n\n\n\n

Page 8.30<\/p>\n\n\n\n

Question 6<\/h4>\n\n\n\n

In the given figure , m||n and \u22202:\u22203=2:3, then find the values of angles denoted by 1,2,…..7,8.<\/strong>
\"\"
Sol:
Let \u22202=2x
\u22203=3x<\/p>\n\n\n\n

\u22202+\u22203=180\u00b0(linear pair)
2x+3x=180\u00b0
5x=180\u00b0
x=\\frac{180^{\\circ}}{5}
x=36\u00b0<\/p>\n\n\n\n

\u22202=2x
=2\u00d736\u00b0
=72\u00b0<\/p>\n\n\n\n

\u22203=3x
=3\u00d736\u00b0
=108\u00b0<\/p>\n\n\n\n

\u22204=\u22202(V.OA)
\u22204=72\u00b0
\u22201=\u22203(V.O.A)
\u22201=108\u00b0<\/p>\n\n\n\n

\u22205=\u22201(corresponding angles)
\u22205=108\u00b0<\/p>\n\n\n\n

\u22207=\u22205(V.O.A)
\u22207=108\u00b0<\/p>\n\n\n\n

\u22206=\u22202(corresponding angles)
\u22206=72\u00b0<\/p>\n\n\n\n

\u22208=\u22206(V.O.A)
\u22208=72\u00b0<\/p>\n\n\n\n

\n\n\n\n

Question 7<\/h4>\n\n\n\n

In figures (i) , (ii) , (iii) , (iv) if AB||CD , then find the value of x and y in each of the following cases:<\/strong>
(i)<\/strong>
\"\"
Sol :
\u2220SRB=\u2220ARP(V.O.A)
x=65\u00b0<\/p>\n\n\n\n

\u2220SRB+\u2220RSD=180\u00b0(Interior angles on same side of transversal are supplementary)
65\u00b0+y=180\u00b0
y=180\u00b0-65\u00b0
y=115\u00b0<\/p>\n\n\n\n

(ii)<\/strong>
\"\"
Sol :
\u2220CSR=\u2220SRB(Alternate interior angle)
x=105\u00b0<\/p>\n\n\n\n

\u2220CSR+\u2220CSQ=180\u00b0
105\u00b0+y=180\u00b0
y=75\u00b0<\/p>\n\n\n\n

(iii)<\/strong>
\"\"<\/p>\n\n\n\n

(iv)<\/strong>
\"\"
Sol :<\/p>\n\n\n\n

\n\n\n\n

Question 8<\/h4>\n\n\n\n

Write the pair of parallel lines in each of the following cases shown in figures below:<\/strong>
(i)<\/strong>
\"\"
Sol :
x+60\u00b0=180\u00b0
x=120\u00b0<\/p>\n\n\n\n

y=60\u00b0(alternate angle)<\/p>\n\n\n\n

(ii)<\/strong>
\"\"
Sol :
x=100\u00b0(alternate angle)<\/p>\n\n\n\n

y=80\u00b0(corresponding angle)<\/p>\n\n\n\n

(iii)<\/strong>
\"\"
Sol :
\u2220D+\u2220A
=115\u00b0+65\u00b0
=180\u00b0<\/p>\n\n\n\n

\u2220A+\u2220B
=115\u00b0+65\u00b0
=180\u00b0<\/p>\n\n\n\n

AD||BC<\/p>\n\n\n\n

\n\n\n\n

Type 3<\/p>\n\n\n\n

Question 9<\/h4>\n\n\n\n

In the given figure , p is a transversal of the lines m and n and \u22201=60\u00b0 and \\angle 2=\\dfrac{2}{3} of a right angle. Prove that m||n.<\/strong>
\"\"
Sol :
Diagram
Given: P is a transversal of the lines m and n
\u22201=60\u00b0 ,\\angle=\\frac{2}{3} of right angle<\/p>\n\n\n\n

To prove: m\u2225n<\/p>\n\n\n\n

Proof: \\angle 2=\\frac{2}{3} of right angle<\/p>\n\n\n\n

\\angle 2=90^{\\circ} \\times \\frac{2}{3}<\/p>\n\n\n\n

\u22202=60\u00b0<\/p>\n\n\n\n

\u22203=\u22202(V.O.A)<\/p>\n\n\n\n

\u22203=60\u00b0<\/p>\n\n\n\n

\u2235\u22201=60\u00b0(\u22201 and \u22203 are pair of corresponding angle)<\/p>\n\n\n\n

If pair of corresponding angles are equal then the lines are parallel<\/p>\n\n\n\n

\u2234 m\u2225n<\/p>\n\n\n\n

\n\n\n\n

Question 10<\/h4>\n\n\n\n

In the given figure , m||n and p||q. If \u22201=75\u00b0 , then prove that \\angle 2=\\angle 1+\\dfrac{1}{3} of right angles<\/strong>
\"\"
Sol :
[Hint: \u2220BAC=\u22201=75\u00b0 [alternate angles]
Now , \u22202+\u2220BAC=180\u00b0  [sum of the interior angle son the same side of transversal n]
\u2234 \u22202=180\u00b0-\u2220BAC
=180\u00b0-75\u00b0
=105\u00b0=75\u00b0+30\u00b0=\\angle 1+\\dfrac{1}{3} of right angle ]<\/p>\n\n\n\n

Sol :
Given: m||n and p|q , \u22201=75\u00b0<\/p>\n\n\n\n

To prove: \u22202=\u22201+\\frac{1}{3} of right angles<\/p>\n\n\n\n

Proof: \u2235p||q (given)
\u2220CDB=\u22201(corresponding angle)
\u2220CDB=75\u00b0
\u2235 m|n (given)<\/p>\n\n\n\n

\u2220CDB+\u22202=180\u00b0
75\u00b0+\u22202=180\u00b0<\/p>\n\n\n\n

\u22202=180\u00b0-75\u00b0
\u22202=105\u00b0<\/p>\n\n\n\n

\u22202=75\u00b0+30\u00b0<\/p>\n\n\n\n

\\angle 2=\\angle 1+\\frac{1}{3} of right angle<\/p>\n\n\n\n

\n\n\n\n

Question 11<\/h4>\n\n\n\n

In the given figure , if x=y and a=b , prove that r||n<\/strong>
\"\"
Sol :
Given:x=y or a=b<\/p>\n\n\n\n

To prove:r||n<\/p>\n\n\n\n

Proof:
x=y(given)<\/p>\n\n\n\n

r||m (\u2235 if corresponding angle of transversal are equal then the given lines is are parallel)<\/p>\n\n\n\n

a=b (given)<\/p>\n\n\n\n

m||n (\u2235 if corresponding angle of transversal are equal then the given lines is are parallel)<\/p>\n\n\n\n

\u2235r||m and m||n<\/p>\n\n\n\n

\u2234r||n<\/p>\n\n\n\n

\n\n\n\n

Question 12<\/h4>\n\n\n\n

lf m,n, p are three lines such that p||m and n\u22a5p, prove that n\u22a5m <\/strong>
Sol :
Diagram
Given: P||m or n\u27c2p<\/p>\n\n\n\n

To prove: n\u27c2m<\/p>\n\n\n\n

Proof: n\u27c2p (given)<\/p>\n\n\n\n

\u22201=90\u00b0<\/p>\n\n\n\n

p||m<\/p>\n\n\n\n

\u22202=\u22201 (alternate interior angle)<\/p>\n\n\n\n

\n\n\n\n

Question 13<\/h4>\n\n\n\n

What are the conditions for two lines to be parallel ?<\/strong>
Sol :
Alternate angles are equal or corresponding angles are equal or , the co-interior angels are supplementary<\/p>\n\n\n\n

\n\n\n\n

Question 14<\/h4>\n\n\n\n

In the given f\u200cigure, l and m are two intersecting lines and p||l and q||m. Prove that p and q are also intersecting lines .<\/strong>
\"\"
Sol:
Given: l and m are intersecting lines p||l and q||m<\/p>\n\n\n\n

Let us assume that p and q are not intersecting lines, which means p||q.<\/p>\n\n\n\n

We know ,<\/p>\n\n\n\n

p||l and q||m<\/p>\n\n\n\n

\u2234It implies that q||p, q||l , l||m [\u2235p||l] [\u2235q||m]<\/p>\n\n\n\n

But , we are given that l and m intersect each other.<\/p>\n\n\n\n

Therefore , our assumption is wrong.<\/p>\n\n\n\n

Hence , p and q are intersecting lines.<\/p>\n\n\n\n

\n\n\n\n

Question 15<\/h4>\n\n\n\n

If a line is perpendicular to any one of the two parallel lines , prove that this line will also be perpendicular to the other parallel line<\/strong>
Sol :
Diagram
Given: p||q and r\u27c2p<\/p>\n\n\n\n

To prove: r\u27c2q<\/p>\n\n\n\n

Proof: p||q (given)<\/p>\n\n\n\n

\u22202=\u22201 (alternate interior angle)<\/p>\n\n\n\n

\u22202=90\u00b0<\/p>\n\n\n\n

\u2234r\u27c2q<\/p>\n\n\n\n

\n\n\n\n

Question 16<\/h4>\n\n\n\n

In the given f\u200cigure, AB||CD and AD||BC, prove that \u2220DAB=\u2220DCB.<\/strong>
\"\"
Sol :
[Hint: AB||DC and transversal BC meets them]
\u2234 \u2220ABC+\u2220BCD=180\u00b0..(i)
Again AD||BC
\u2234 \u2220DAB+\u2220ABC=180\u00b0..(ii)
From (i) and (ii) , \u2220ABC+\u2220BCD=\u2220DAB+\u2220ABC=180\u00b0
or \u2220DAB=\u2220BCD=\u2220DAB ]
Sol :
Given: AB||CD and AD||BC<\/p>\n\n\n\n

To prove: \u2220DAB=\u2220DCB<\/p>\n\n\n\n

Proof: AB||CD (given)<\/p>\n\n\n\n

\u2220ABC+\u2220DCB=180\u00b0..(i)<\/p>\n\n\n\n

AD||BC(given)<\/p>\n\n\n\n

\u2220DAB+\u2220ABC=180\u00b0..(ii)<\/p>\n\n\n\n

From equation (i) and (ii)<\/p>\n\n\n\n

\u2220DAB+\u2220ABC=\u2220ABC+\u2220DCB<\/p>\n\n\n\n

\u2220DAB=\u2220DCB<\/p>\n\n\n\n

\n\n\n\n

Question 17<\/h4>\n\n\n\n

A transversal intersects two given lines m and n in such a way that interior angles on the same side of the transversal are equal , then it is always true that the given lines are parallel ? If not , state the condition under which the two lines will be parallel.<\/strong>
[Hint: m||n only when transversal line is perpendicular to both m and n]
Sol :
Diagram
\u2220AOC+\u2220BOC=180\u00b0
6x+30\u00b0+4x=180\u00b0
10x+30\u00b0=180\u00b0
10x=180\u00b0-30\u00b0
10x=150\u00b0
x=\\frac{150^{\\circ}}{10}
x=15\u00b0<\/p>\n\n\n\n

\n\n\n\n

Question 18<\/h4>\n\n\n\n

Prove that the two lines that are respectively perpendicular to two intersecting lines intersect each other .<\/strong>
\"\"
[Hint: As shown in the given figure , lines MN and PQ are perpendicular to intersecting lines AB and CD respectively. If possible , suppose lines MN and PQ do not intersect , then definitely MN||PQ.
But we know that two parallel lines cannot be simultaneously perpendicular to two intersecting lines. So , it is wrong to assume that MN||PQ. Hence MN and PQ are intersecting lines ]
Sol :<\/p>\n\n\n\n

\"\"\/<\/a><\/figure>\n\n\n\n
\n\n\n\n

Question 19<\/h4>\n\n\n\n

In the given f\u200cigure, lines AB and CD are parallel and any point P lies between these lines . Prove that \u2220ABP+\u2220CDO=\u2220DPB<\/strong>
\"\"
[Hint: Through point P , draw a line PM||AB and proceed.]
Sol :<\/p>\n\n\n\n

\"\"\/<\/a><\/figure>\n\n\n\n
\n\n\n\n

Question 20<\/h4>\n\n\n\n

If a transversal intersects two lines such that the bisectors of a pair of alternate angles are parallel, then prove that the two lines are parallel<\/strong>
Sol :
[Hint: Proceed as in solved example 20 page 8.27]
Diagram
Given: \u2220AOB=\u2220COD..(i)
\u2220BOC=\u2220DOA..(ii)<\/p>\n\n\n\n

from diagram
\u2220AOB+\u2220BOC+\u2220COD+\u2220DOA=360\u00b0
\u21d22\u2220COD+2\u2220DOA=360\u00b0
\u21d2\u2220COD+\u2220DOA=\\frac{360^{\\circ}}{2}
\u21d2\u2220AOC=180\u00b0<\/p>\n\n\n\n

Similarly \u2220BOD=180\u00b0<\/p>\n\n\n\n

Sum of all angle at a point on a straight line is 180\u00b0<\/p>\n\n\n\n

\n
KC Sinha Solutions<\/a><\/div>\n\n\n\n
KC Sinha Class 9 Solutions<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Exercise 8.2 Type 1 Question 1  Answer the following questions :(i) If any transversal intersects two parallel lines in such a way that the ratio of interior angles on the same side is 2 : 7 , then f\u200cind the measure of the bigger angle in degrees.(ii) If two parallel lines are intersected by a […]<\/p>\n","protected":false},"author":302,"featured_media":623859,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"newspack_featured_image_position":"","newspack_post_subtitle":"","newspack_article_summary_title":"Overview:","newspack_article_summary":"","newspack_hide_updated_date":false,"newspack_show_updated_date":false,"footnotes":""},"categories":[921],"tags":[],"boards":[],"class_list":["post-623860","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-9","entry"],"yoast_head":"\nKC Sinha: Exercise 8.2 - Mathematics Solution Class 9 Chapter 8 Lines and Angles - IndCareer Schools<\/title>\n<meta name=\"description\" content=\"Exercise 8.2 Type 1 Question 1  Answer the following questions :(i) If any transversal intersects two parallel lines in such a way that the ratio of\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-2-mathematics-solution-class-9-chapter-8-lines-and-angles\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"KC Sinha: Exercise 8.2 - 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