Q.1Â Â Â Â In figure sides QP and RQ of PQR are produced to points S and T respectively. If 135Âº and 110Âº, find
We have ,
180Âº [Linear pair]
+ 135Âº = 180Âº
180Âº â€“ 135Âº = 45Âº
Now , [By exterior angle theorem]
110Âº = 45Âº +
= 110Âº â€“ 45Âº = 65Âº
Hence = 65Âº
Q.2Â Â Â Â Â In figure, 62Âº, 54Âº. If YO and ZO are the bisectors of respectively of XYZ, find
Consider XYZ,
= 180Âº [Angle - sum property]
62Âº + 54Âº + = 180Âº [Since 62Âº , 54Âº]
Since YO and ZO are bisectors of
Therefore
and ,
In OYZ , we have
180Âº [Angle sum property]
27Âº + 32Âº = 180Âº
= 180Âº â€“ 27Âº â€“ 32Âº
= 180Âº â€“ 59Âº = 121Âº
Hence, 32Âº
and = 121Âº
Q.3Â Â Â Â Â In figure if AB || DE, = 35Âº and 53Âº, find .
Since AB || DE and transversal AE intersects them at A and E respectively.
Therefore [Alternate angles]
35Âº [Since 35Âº]
In DEC, we have
180Âº [Angle sum property]
35Âº + 53Âº = 180Âº
180Âº â€“ 35Âº â€“ 53Âº
= 180Âº â€“ 88Âº = 92Âº
Hence , 92Âº
Q.4Â Â Â Â Â In figure, if lines PQ and RS intersect at point T, such that 40Âº, 95Âº and 75Âº, find .
In PRT, we have
180Âº [Angle - sum property]
40Âº + 95Âº = 180Âº
180Âº â€“ 40Âº â€“ 95Âº
= 180Âº â€“ 135Âº = 45Âº
[Vertically opp. angles]
45Âº [Since RTP = 45Âº (proved)]
In TQS we have
180Âº [Angle - sum property]
45Âº + 75Âº = 180Âº [SinceÂ (proved) and ]
180Âº â€“ 45Âº â€“ 75Âº
= 180Âº â€“ 120Âº = 60Âº
Hence , 60Âº
Q.5Â Â Â Â Â In figure if PQ PS, PQ|| SR, = 28Âº and = 65Âº, then find the values of x and y.
Using exterior angle property in SRQ, we have
65Âº = 28Âº + [Since 65Âº , 28Âº]
QSR = 65Âº â€“ 28Âº = 37Âº
Since PQ|| SR and the transversal PS intersects them at P and S respectively.
Therefore 180Âº [Sum of consecutive interior angles is 180Âº]
90Âº = 180Âº
y + 37Âº + 90Âº = 180Âº
y = 180Âº â€“ 90Âº â€“ 37Âº
= 180Âº â€“ 127Âº = 53Âº
In the right , we have
90Âº
x + 53Âº = 90Âº
x = 90Âº â€“ 53Âº = 37Âº
Hence, x = 37Âº
and y = 53Âº
Q.6Â Â Â Â In figure , the side QR of PQR is produced to a point S. If the bisectors of meet at point T, then prove that .
In we have ext.
Â Â ( By exterior angle theorem )
... (1)
[Since QT and RT are bisectors of respectively therefore ]
In QRT. we have ext. ... (2)
From (1) and (2) , we get
[ Since ]
Hence ,
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