Question

"Question 5 In the given figure, if PQ ⊥ PS, PQ || SR, ∠SQR = 28º and ∠QRT = 65º, then find the values of x and y.

Class 9 - Math - Lines and Angles Page 108"

Answer 1

Exterior Angle of a triangle:

If a side of a triangle is produced then the exterior angle so formed is equal to the sum of the two interior opposite angles.

Interior angles on the same side of the transversal:

 

The pair of interior angles on the same side of the transversal are called consecutive interior angles or allied angles or co interior angles.

 

If a transversal intersects two Parallel Lines then each pair of interior angles on the same side of the  transversal is supplementary(180°)

 

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Solution:

 

Given:

PQ⊥SP, PQ||SR, ∠SQR=28° & ∠QRT=65°

In, ∆QSR,

 ∠QRT=∠RQS+∠QSR

 (exterior angle is equal to sum of the two opposite interior angles)

⇒65∘=28∘+∠QSR 

⇒∠QSR=65∘−28∘=37∘

PQ⊥SP     (given)

∠QPS=90°

 

∠QPS+∠PSR=180∘    

 (the sum of consecutive interior angles on the same side of the transversal in 180∘ )

 

⇒90∘+∠PSR=180∘ 

⇒∠PSR=180∘−90∘=90∘ 

⇒∠PSQ+∠QSR=90∘

 ⇒y+37∘=90∘ 

⇒Y=90∘−37∘=53∘

 

In △PSQ,

∠PSQ+∠QSP+∠QPS=180∘   

 

[sum of interior angles of a ∆ is 180∘.]

 

⇒x+y+90∘=180∘ 

x+53∘+90∘=180∘ 

x+143∘=180∘ 

x=180∘−143∘=37∘

 

Hence,x= 37° & y=53°

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Answer 2

Hope this helps u.....