# Lines and Angles : Exercise 6.1 (Mathematics NCERT Class 9th)

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**Q.1Â Â Â In figure , lines AB and CD intersect at O. If and , find and reflex .**

Since OA and OB are opposite rays. Therefore AB is a line.

Since ray OC stands on AB. Therefore,

[Linear Pairs]

[Since ]

[Since (Given)]

Therefore Reflex

Since OC and OD are opposite rays. Therefore CD is a line.

Since ray OE stands on CD. Therefore -

[Linear Pairs]

[Since ]

[Since (Given)]

Hence,

and

**Q.2Â Â Â Â Â Â In figure , lines XY and MN intersect at O. If and a : b = 2 : 3, find c.**

Since a : b = 2 : 3 and a + b = = 90Âº and sum of ratios = 2 + 3 = 5

Therefore

and

Since OM and ON are opposite rays. Therefore MN is a line.

Since ray OX stands on MN. Therefore,

[Linear Pairs]

Hence ,

**Q.3Â Â Â Â Â Â In figure , then prove that .**

Since QS and QR are opposite rays. Therefore, SR is a line.

Since QP stands on the line SR.

Therefore [Linear Pair] ...(1)

Again RQ and RT are opposite rays. Therefore, QT is a line.

Since PR stands on the line QT.

Therefore [Linear Pair] ... (2)

From (1) and (2), we have

[Since Each side = 180Âº] ... (3)

Also Â Â Â Â Â Â (given)Â Â Â Â Â ................(4)

Subtracting (4) from (3), we have

**Q.4Â Â Â Â Â Â In figure, if x + y = w + z, then prove that AOB is a line.**

We know that the sum of all angles around a point is equal to Â 360Âº

Therefore

But [Given]

ThereforeÂ

Thus , form linear pairs. Consequently OA and OB are two opposite rays.

Therefore AOB is a straight line.

**Q.5Â Â Â Â Â Â In figure POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that **

Since OR is perpendicular to the line PQ.

Therefore [Since Each = 90Âº]

**Q.6Â Â Â Â Â It is given that and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects and reflex .
**

**Sol.**Since XY is produced to point P. Therefore XP is a straight line.

Since YZ stands on XP.

Therefore [Linear Pair]

[Since ]

Since ray YQ bisects

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