NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles

Q. In the given figure, if lines PQ and RS intersect at a point T such that ∠PRT = 40°, ∠RPT = 95° and ∠TSQ = 75°. Find ∠SQT.

ntersect

Ans. Given : ∠P = 95°, ∠R = 40° and ∠S = 75°
In DPRT,
∠P + ∠R + ∠PTR = 180°
⇒ ∠PTR = 180° – ∠P – ∠R
⇒ ∠PTR = 180° – 95° – 40°
⇒ ∠PTR = 45°
... ∠STQ = ∠PTR = 45°
(Vertically opposite angle)
In DSTQ,
∠Q + ∠S + ∠STQ = 180°
⇒ ∠SQT + 75° + 45° = 180°
⇒ ∠SQT = 180° – 120°
= 60°
∠SQT = 60°

Q. In the given figure, POQ is a line. Ray OR is perpendicular to PQ. OS is another ray lying between rays OP and OR. Prove that ∠ROS = 1/2
(∠QOS – ∠POS).

perpendicular

Ans. $$\\ Given : PQ\bot OR\\ To prove : ∠ROS =\frac{1}{2}(∠QOS – ∠POS)\\ Proof : ∠QOS = ∠ROS + ∠QOR\\ ∠POS = ∠POR – ∠ROS\\ \frac{(–)(–)+}{∠QOS – ∠POS = ∠QOR – ∠POR + 2 ∠ROS}\\ ⇒ ∠QOS – ∠POS = 90° – 90° + 2 ∠ROS\\ \ ∠ROS =\frac{1}{2} (∠QOS – ∠POS)$$

Q. If two lines are intersect, then the vertically opposite angles are equal.

Ans. If two lines are intersect, then the vertically opposite angles are equal.

Q. In the given figure, PQ ⊥ PS, PQ || SR, ∠SQR = 28° and ∠QRT = 65°, then find the value of x and y.

perpendiculars