EXERCISE 6.2
Lines & Angles Theorem 6.7 Page No 105, CBSE Class 9th
Q 1, Ex 6.2, Page No 103, Lines And Angles, CBSE Class 9th
Page No 103:
Question 1:
In the given figure, find the values of x and y and then show that AB || CD.
Answer:
It can be observed that,50º + x = 180º (Linear pair)
x = 130º … (1)
Also, y = 130º (Vertically opposite angles)
As x and y are alternate interior angles for lines AB and CD and also measures of these angles are equal to each other, therefore, line AB || CD.
Q 2, Ex 6.2, Page No 104, Lines And Angles, Class 9th Mathematics
Page No 104:
Question 2:
In the given figure, if AB || CD, CD || EF and y: z = 3: 7, find x.
Answer:
It is given that AB || CD and CD || EF∴ AB || CD || EF (Lines parallel to the same line are parallel to each other)
It can be observed that
x = z (Alternate interior angles) … (1)
It is given that y: z = 3: 7
Let the common ratio between y and z be a.
∴ y = 3a and z = 7a
Also, x + y = 180º (Co-interior angles on the same side of the transversal)
z + y = 180º [Using equation (1)]
7a + 3a = 180º
10a = 180º
a = 18º
∴ x = 7a = 7 × 18º = 126º
Q 3, Ex 6.2, Page No 104, Lines And Angles, Class 9th Mathematics
Question 3:
In the given figure, If AB || CD, EF ⊥ CD and ∠GED = 126º, find ∠AGE, ∠GEF and ∠FGE.
Answer:
It is given that,AB || CD
EF ⊥ CD
∠GED = 126º
⇒ ∠GEF + ∠FED = 126º
⇒ ∠GEF + 90º = 126º
⇒ ∠GEF = 36º
∠AGE and ∠GED are alternate interior angles.
⇒ ∠AGE = ∠GED = 126º
However, ∠AGE + ∠FGE = 180º (Linear pair)
⇒ 126º + ∠FGE = 180º
⇒ ∠FGE = 180º − 126º = 54º
∴ ∠AGE = 126º, ∠GEF = 36º, ∠FGE = 54º
Q 4, Ex 6.2, Page No 104, Lines And Angles, Class 9th Maths
Question 4:
In the given figure, if PQ || ST, ∠PQR = 110º and ∠RST = 130º, find ∠QRS.[Hint: Draw a line parallel to ST through point R.]
Answer:
∠PQR + ∠QRX = 180º (Co-interior angles on the same side of transversal QR)
⇒ 110º + ∠QRX = 180º
⇒ ∠QRX = 70º
Also,
∠RST + ∠SRY = 180º (Co-interior angles on the same side of transversal SR)
130º + ∠SRY = 180º
∠SRY = 50º
XY is a straight line. RQ and RS stand on it.
∴ ∠QRX + ∠QRS + ∠SRY = 180º
70º + ∠QRS + 50º = 180º
∠QRS = 180º − 120º = 60º
Q 5, Ex. 6.2, Page No 104, Lines & Angles, Maths Class 9th
Question 5:
In the given figure, if AB || CD, ∠APQ = 50º and ∠PRD = 127º, find x and y.
Answer:
∠APR = ∠PRD (Alternate interior angles)50º + y = 127º
y = 127º − 50º
y = 77º
Also, ∠APQ = ∠PQR (Alternate interior angles)
50º = x
∴ x = 50º and y = 77º
Q 6, Ex. 6.2, Page No 104, Lines & Angles, Class 9th Maths
Question 6:
In the given figure, 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.
Answer:
Let us draw BM ⊥ PQ and CN ⊥ RS.
As PQ || RS,
Therefore, BM || CN
Thus, BM and CN are two parallel lines and a transversal line BC cuts them at B and C respectively.
∴∠2 = ∠3 (Alternate interior angles)
However, ∠1 = ∠2 and ∠3 = ∠4 (By laws of reflection)
∴ ∠1 = ∠2 = ∠3 = ∠4
Also, ∠1 + ∠2 = ∠3 + ∠4
∠ABC = ∠DCB
However, these are alternate interior angles.
∴ AB || CD
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