# NCERT Solutions for Class 9 Maths Unit 6

## Lines and Angles Class 9

### Unit 6 Lines and Angles Exercise 6.1, 6.2, 6.3 Solutions

Line
A line is a collection of points that extends without limit in a straight formation. A line can be named by a single letter, like line l, or it can be named according to two points that it contains, like line AB. The second way of naming a line indicates an important property common to all lines: any two points in space determine a line.

Angles
Technically speaking, an angle is the union of two rays (lines that extend infinitely in just one direction) that share an endpoint (called the vertex of the angle). The measure of an angle is how far you must rotate one of the rays such that it coincides with the other.

### Exercise 6.1 : Solutions of Questions on Page Number : 96

Q1 :

In the given figure, lines AB and CD intersect at O. If and find ∠BOE and reflex ∠COE.

Q2 :

In the given figure, lines XY and MN intersect at O. If ∠POY = and a:b = 2 : 3, find c.

Let the common ratio between a and b be x.

a = 2x, and b = 3x

XY is a straight line, rays OM and OP stand on it.

∴ ∠ XOM + ∠ MOP + ∠ POY = 180º

b + a + ∠ POY = 180º

3x + 2x + 90º = 180º

5x = 90º

x = 18º

a = 2x = 2 x 18 = 36º

b = 3x= 3 x 18 = 54º

MN is a straight line. Ray OX stands on it.

b + c = 180º (Linear Pair)

54º + c = 180º

c = 180º - 54º = 126º

c = 126º

Q3 :

In the given figure, ∠ PQR = ∠ PRQ, then prove that ∠ PQS = ∠ PRT.

Q4 :

In the given figure, if then prove that AOB is a line.

Q5 :

In the given figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that

Q6 :

It is given that ∠XYZ = and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects ∠ZYP, find ∠XYQ and reflex ∠QYP.

### Exercise 6.2 : Solutions of Questions on Page Number : 103

Q1 :

In the given figure, find the values of x and y and then show that AB || CD.

Q2 :

In the given figure, if AB || CD, CD || EF and y: z = 3: 7, find x.

Q3 :

In the given figure, If AB || CD, EF ⊥ CD and ∠ GED = 126º, find ∠ AGE, ∠ GEF and ∠ FGE.

Q4 :

In the given figure, if PQ || ST, ∠ PQR = 110º and ∠ RST = 130º, find ∠ QRS.

[Hint: Draw a line parallel to ST through point R.]

Q5 :

In the given figure, if AB || CD, ∠ APQ = 50º and ∠ PRD = 127º, find x and y.

Q6 :

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.

### Exercise 6.3 : Solutions of Questions on Page Number : 107

Q1 :

In the given figure, sides QP and RQ of ΔPQR are produced to points S and T respectively. If ∠ SPR = 135º and ∠ PQT = 110º, find ∠ PRQ.

Q2 :

In the given figure, ∠ X = 62º, ∠ XYZ = 54º. If YO and ZO are the bisectors of ∠ XYZ and ∠ XZY respectively of ΔXYZ, find ∠ OZY and ∠ YOZ.

Q3 :

In the given figure, if AB || DE, ∠ BAC = 35º and ∠ CDE = 53º, find ∠ DCE.

Q4 :

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

Q5 :

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

Q6 :

In the given figure, the side QR of ΔPQR is produced to a point S. If the bisectors of ∠PQR and ∠PRS meet at point T, then prove that ∠QTR=∠QPR.