KSEEB Solutions for Class 9 Maths Chapter 3 Lines and Angles Ex 3.2

KSEEB Solutions for Class 9 Maths Chapter 3 Lines and Angles Ex 3.2 are part of KSEEB Solutions for Class 9 Maths. Here we have given Karnataka Board Class 9 Maths Chapter 3 Lines and Angles Exercise 3.2.

Karnataka Board Class 9 Maths Chapter 3 Lines and Angles Ex 3.2

Question 1.
In Fig. 3.28. find the values of x and y and then shows that AB||CD.
KSEEB Solutions for Class 9 Maths Chapter 3 Lines and Angles Ex 3.2 1
Answer:
i) ∠x = ?
ii) ∠y = ?
iii) Show tht AB||CD.
KSEEB Solutions for Class 9 Maths Chapter 3 Lines and Angles Ex 3.2 2
(i) PQ Bisects AB at R.
∠PRA and ∠ARS are on straight line PQ.
∴ ∠PRA + ∠ARS = 180° Adjacent angles
50 + ∠ARS = 180°
∠ARS = 180 – 50
∴ x = ∠ARS = 130° (i)

(ii) Similarly, PQ straight line intersects straight line CD at S.
∴ ∠CSQ = ∠RSD . Vertically opposite angles
∴ y = ∠RSD = 130° (ii)
From (i) and (ii),
∠ARS = ∠RYD
x = y = 130°

(iii) Therse are mutual alternate angles
∴ x = y = 130°
∴ AB || CD.

Question 2.
In Fig. 3.29, if AB || CD, CD || EF and y : z = 3 : 7, find x.
KSEEB Solutions for Class 9 Maths Chapter 3 Lines and Angles Ex 3.2 3
Answer:
PQ straight line intersects AB||CD]|EF at points R, S, T.
If y : z = 3 : 7, then value of ‘x’ = ?
PQ straight line intersects CD||EF at S and T.
∴ ∠CSR = ∠DST = y (∵ vertically opposite angle)
But, ∠DST + ∠STF = 180°
(∵ Sum of interior angles in the same side)
∴ y + z = 180°
But, y : z = 3 : 7.
Sum of ratio 3 + 7 = 10
If ratio 10 means 180
If ratio 9 means ?
\(\frac{180 \times 3}{10}=54^{\circ}\)
∴ y = 54°
y + z = 180
54 + z = 180
∴ z = 180 – 54
∴ z = 126°
But, x + y = 180
(∵ Sum of interior angles on one side)
x + 54 = 180
∵ x = 180 – 54
∵ x = 126°.

Question 3.
In Fig. 3.30, If AB||CD, EF ⊥ CD and ∠GED = 126°, find ∠AGE, ∠GEF and ∠FGE.
KSEEB Solutions for Class 9 Maths Chapter 3 Lines and Angles Ex 3.2 4
Answer:
AB||CD and EF⊥CD and ∠GED = 126°
KSEEB Solutions for Class 9 Maths Chapter 3 Lines and Angles Ex 3.2 5
∠AGE =?
∠GEF =?
∠FGE =?
∠GED = 126°.
∴ ∠GEF + ∠ZFED = 126°
∠GEF + 90° = 126° ∵ EF ⊥ CD.
∴ GEF = 126-90
∴  GEF = 36°.
∴ ∠GFE = 90° . EF ⊥ AB
∴ In ∆GEF
∠FGE + ∠GFE + ∠GEF = 180°
∠FGE + 90° + 36° = 180°
∠FGE + 126° = 180°
∴ ∠FGE = 180-126
∴ ∠FGE = 54°.
∠AGE + ∠FGE = 180° (Adjacent angles)
∠AGE + 54° = 180°
∴ ∠AGE =180-54
∠AGE = 126°
∴ ∠AGE = 126°
∠GEF = 36°
∠FGE = 54°.

Question 4.
In Fig. 3.31, If PQ||ST, ∠PQR =110°, and ∠RST = 130°, find ∠QRS.
(Hint: Draw a line parallel to ST through point R).
KSEEB Solutions for Class 9 Maths Chapter 3 Lines and Angles Ex 3.2 6
Answer:
KSEEB Solutions for Class 9 Maths Chapter 3 Lines and Angles Ex 3.2 7
Data: PQ||ST and ∠PQR= 110° and ∠RST= 130°.
To Prove: ∠QRS =?
Construction: Draw ST||UV through ‘R’
Proof: PQ||ST (Data)
ST || UV (Construction)
∴ PQ||ST||UV
PQ||UV.
∴ ∠PQR + ∠URQ = 180° (Sum of interior angles)
110 + ∠URQ = 180°
∠URQ = 180- 110
∴ ∠URQ = 70° (i)
Similarly, ST||UV.
∴ ∠RST + ∠SRV = 180°
130 + ∠SRV = 180
∴ ∠SRV = 180 – 130
∴ ∠SRV = 50° (ii)
But, URV is a straight line.
∴ ∠URV = 180°.
∠URQ + ∠QRS + ∠RSV = 180°
70 + ∠QRS + 50 = 180
∠QRS + 120 = 180°
∴ ∠QRS = 180 – 120
∴ ∠QRS = 60°.

Question 5.
In Fig. 3.32. if AB||CD, ∠APQ = 50° and ∠PRD = 127°, find x and y.
KSEEB Solutions for Class 9 Maths Chapter 3 Lines and Angles Ex 3.2 8
Answer:
If AB || CD and ∠APQ= 50° and ∠PRD = 127°, then ∠x = ? ∠y = ?
AB || CD PQ straight line intersects these at P and Q.
∴ ∠APQ = ∠PQR . alternate angles.
50 = ∠PQR = x
∴ x = 50°
Similarly, AB || CD PR straight line intersects these at P and R.
∴ ∠APR = ∠PRD
∠APQ + ∠QPR = ∠PRD
50 + ∠QPR = 127
∠QPR = 127 – 50
∴∠QPR = 77 °
∠QPR = y = 77 °
∴ x = 50 °,
y = 77 °.

Question 6.
In Fig. 3.33, PQ and RS are two mirrors placed parallel to each other. An incident ray AB strikes the mirror PQ at B, the reflected ray moves along the path BC and strikes the mirror RS at C and again reflects back along CD.
Prove that AB||CD.
KSEEB Solutions for Class 9 Maths Chapter 3 Lines and Angles Ex 3.2 9
Answer:
Data: PQ and RS are two mirrors placed parallel to each other. An incident ray AB strikes the mirror PQ at B. the reflected ray moves along the path BC and strikes the mirror RS at C and again reflects back along CD.
To Prove: AB || CD.
Proof: PQ is a mirror.
AB is an incident ray, BC is a reflected ray. ∠ABC is the angle of incidence.
∠BCD is an angle of reflection.
The incident ray formed in the PQ mirror is equal to the reflected ray formed in the RS mirror.
∴ ∠ABC = ∠BCD.
These are pair of alternate interior angles, then AB || CD.

We hope the KSEEB Solutions for Class 9 Maths Chapter 3 Lines and Angles Ex 3.2 help you. If you have any query regarding Karnataka Board Class 9 Maths Chapter 3 Lines and Angles Exercise 3.2, drop a comment below and we will get back to you at the earliest.

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