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NCERT Solutions for Class 9 Mathematics Chapter 6 Lines And Angles

NCERT Solution For Class 9 Mathematics

Lines And Angles Here is the CBSE Mathematics Chapter 6 for Class 9 students. Summary and detailed explanation of the lesson, including the definitions of difficult words. All of the exercises and questions and answers from the lesson's back end have been completed. NCERT Solutions for Class 9 Mathematics Lines And Angles Chapter 6 NCERT Solutions for Class 9 Mathematics Lines And Angles Chapter 6 The following is a summary in Hindi and English for the academic year 2023-24. You can save these solutions to your computer or use the Class 9 Mathematics.

Question 1

NCERT Solutions for Class 9 Mathematics Chapter 6 Lines And Angles

NCERT Solution For Class 9 Mathematics

Ray OE bisects ∠AOB and OF is the ray opposite to OE. Show that ∠FOB = ∠FOA.

Rays OA, OB. OC, OD and OE have the common initial point O. Show that ∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA = 360°.

Prove “if two lines intersect each other, then the vertically opposite angles are equal.”

In figure, OP bisects ∠AOC, OQ bisects ∠BOC and OP ⊥ OQ. Show that the points A, O and B are collinear.

In figure, if y = 20°, prove that the line AOB is a straight line.

In the given figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex angle EOC.

In figure, lines PQ and RS intersect each other at point O. If ∠POR : ∠ROQ = 5 : 7, find all the angles.

In figure, ray OS stands on a line POQ. Ray OR and ray OT are angle bisectors of ∠POS and ∠SOQ, respectively. If ∠POS = x, find ∠ROT.

If the complement of an angle is one-third of its supplement, find the angle.

In figure, find the value of x.

In figure, if ∠POR and ∠QOR form a linear pair and a - b = 80° then find the value of a and b.

What value of x would make XOY a line if ∠XOZ = 7x + 20° and ∠YOZ = 3x in figure.

An angle is of its supplement. Find the angle.

Two supplementary angles are in the ratio 4 : 5. Find the angles.

In figure, lines AB and CD intersect at O. If ∠AOD : ∠DOC = 4 : 5 then find ∠COB.

An angle is equal to five times of its complement. Find the measure of the angle.

In figure, lines I and m intersect each other at O. If x = 40°, then, find the values of y, z and w.

If OA, OB. OC and OD are the rays such that ∠AOB = ∠COD = 150°, ∠BOC = 30° and ∠AOD = 30°. Is it true that AOC and BOD are straight lines? Justify your answer.

In figure, angles x and y are supplementary angles. If ∠x = 110°, find the value of y.

In figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.

In figure, lines XY and MN intersect at O. If ∠POY = 90° and a : b = 2 : 3, find c.

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

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

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

It is given that ∠XYZ = 64° and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects ∠ZYP, find ∠XYQ and reflex ∠QYP.

In figure, if AOB is a line, OP bisects ∠BOC and OQ bisects ∠AOC, show that ∠POQ is a right angle.

In the given figure, lines XY and MN intersect each other at O. If ∠POY = 90° and a : b = 2 : 3, then find c.

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

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

In figure, if AB || CD, EF ⊥ CD and ∠GED = 126°, find ∠AGE, ∠GEF and ∠FGE.

In figure, if PQ || ST, ∠PQR = 110° and ∠RST = 130°, find ∠QRS.[Hint. Draw a line parallel to ST through point R.]

In figure, if AB II CD, ∠APQ = 50° and ∠PRD = 127°, find x and y.

In 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.

In figure, if x = y and a = b, prove that r || n.

If two lines are perpendicular to the same line, prove that they are parallel to each other.

In figure, EF is a transversal to two parallel lines AB and CD, GM and HL are the bisectors of the corresponding angles EGB and EHD. Prove that GM || HL. [Hint. First prove that ∠EGM = ∠GHL]

In figure, if AB || CD then find the value of y.

In figure, is line I parallel to m? Justify your answer.

In figure, PQ || RS and T is any point as shown in the figure then show that ∠PQT + ∠QTS + ∠RST = 360°.

If l, m, n are three lines such that l || m and n ⊥ l, then prove that n ⊥ m.

In figure, l || m, show that ∠1 + ∠2 - ∠3 = 180°.

In figure, AB || CD and CD || EF. Also EA ⊥ AB. If ∠BEF = 40°, then find x, y, z.

In the figure below, l1 || l2 and a1 || a2. Find the value of x.

Two parallel lines are intersected by a transversal. Then, prove that the bisectors of two pairs of interior angles enclose a rectangle.

In the figure, if PQ || RS, ∠MXQ = 135° and ∠MYR = 40°, find ∠XMY.

If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.

In the figure, AB || CD and CD || EF. Also EA ⊥ AB. If ∠BEF = 55°, find the values of x, y and z.

In figure, if AB || CD then find the value of x.

AB and CD are the bisectors of the two alternate interior angles formed by the intersection of a transversal 't' with parallel lines l and m. Show that AB || CD.

In figure, l || m. ∠1 and ∠2 are in the ratio 5 : 4. Find ∠3 and ∠4.

In figure, PQ || RS. Find x.

In figure, l1 || l2 and m1 || m2. Prove that ∠1 + ∠2 = 180°.

In figure, if PQ || RS and ∠PXM = 50° and ∠MYS = 120°, find the value of x.

In figure, find the value of ∠QRP when QP || RT.

In figure, if AB II CD, then find ∠PQR.

In figure, prove that AB || EF.

In figure, if AB || DE then find the measure of ∠BCD.

In figure, prove that AB || EF.

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

In figure, PQ || RS, ∠PAB = 80° and ∠ACS = 120°. Find the value of x.

In figure, sides QP and RQ of ∆PQR are produced to points S and T respectively. If ∠SPR = 135° and ∠PQT = 110°. find ∠PRQ.

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

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

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

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

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

Prove that if one angle of a triangle is equal to the sum of the other two angles, the triangle is right angled.

In figure, the bisectors of ∠ABC and ∠BCA intersect each other at the point O. Prove that

The side EF, FD and DE of a triangle DEF are produced in order forming three exterior angles DFP, EDQ and FER respectively. Prove that∠DFP + ∠EDQ + ∠FER = 360°. OR Prove that the sum of the exterior angles of a triangl is 360°.

In the figure, AD and CE are the angle bisectors of ∠A and ∠C respectively. If ∠ABC = 90° then find ∠AOC.

In figure, find the value of x.

In figure, if QT ⊥ PR, ∠TQR = 40° and ∠SPR = 30°, find the value of x and y.

In figure, if line segment AB intersects CD at O such that ∠OAD = 80°, ∠ODA = 50° and ∠OCB = 40°, then find ∠OBC.

In figure, AP and DP are bisectors of two adjacent angles A and D of a quadrilateral ABCD. Prove that 2 ∠APD = ∠B + ∠C.

In figure, AB II DE, ∠ABC = 75°, ∠CDE = 145°, find ∠BCD.

In figure, the side BC of ∆ABC is produced to D. The bisector of ∠A meets BC in L. Prove that ∠ABC + ∠ACD = 2 ∠ALC

Prove that the sum of the angles of a hexagon is 720°.

In figure, PS is the bisector of ∠QPR and PT ⊥ QR. Show that

An angle is $open parentheses 1 fifth close parentheses to the power of t h end exponent$ of its supplement. Find the angle.

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

$angle ROS equals 1 half left parenthesis angle QOS minus angle POS right parenthesis$

In figure, if PQ || ST, ∠PQR = 110° and ∠RST = 130°, find ∠QRS.
[Hint. Draw a line parallel to ST through point R.]

In figure, EF is a transversal to two parallel lines AB and CD, GM and HL are the bisectors of the corresponding angles EGB and EHD. Prove that GM || HL.

[Hint. First prove that ∠EGM = ∠GHL]

In figure, PQ || RS and T is any point as shown in the figure then show that

∠PQT + ∠QTS + ∠RST = 360°.

In the figure below, l₁ || l₂ and a₁ || a₂. Find the value of x.

In figure, l₁ || l₂ and m₁ || m₂. Prove that ∠1 + ∠2 = 180°.

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

$angle QTR equals 1 half angle QPR$

In figure, the bisectors of ∠ABC and ∠BCA intersect each other at the point O. Prove that $angle BOC equals 90 degree plus 1 half angle straight A$

The side EF, FD and DE of a triangle DEF are produced in order forming three exterior angles DFP, EDQ and FER respectively. Prove that
∠DFP + ∠EDQ + ∠FER = 360°.

OR

Prove that the sum of the exterior angles of a triangl is 360°.

In figure, PS is the bisector of ∠QPR and PT ⊥ QR. Show that $angle T P S equals 1 half left parenthesis angle straight Q minus angle straight R right parenthesis.$

The degree measures of three angles of a triangle are x, y and z. If $straight z equals fraction numerator straight x plus straight y over denominator 2 end fraction comma$ then find the value of z.

In figure, the sides AB and AC of ∆ABC are produced to points E and D respectively. If bisectors BO and CO of ∠CBE and ∠BCD respectively meet at point O, then prove that $angle BOC equals 90 degree minus 1 half angle BAC.$

An acute angle
measures between 0° and 90°
is exactly equal to 90°
is greater than 90° but less than 180°
is equal to 180°

An angle which measures between 0° and 90° is called
a straight angle

an acute angle

a right angle
an obtuse angle

A right angle
measures between 0° and 90°
is exactly equal to 90°
is greater than 90° but less than 180°
is equal to 180°

An angle which is exactly equal to 90° is called
an obtuse angle
an acute angle
a straight angle
a right angle

An angle which is greater than 90° and less than 180° is called
a right angle
a straight angle

an acute angle

an obtuse angle

An obtuse angle
measures between 0° and 90°
is greater than 90° but less than 180°
is exactly equal to 90°
is exactly equal to 180°

A straight angle
is greater than 90° but less than 180°
is exactly equal to 90°
measures between 0° and 90°
is exactly equal to 180°

An angle which is exactly equal to 180° is called
a right angle
a straight angle
an acute angle
an obtuse angle

A reflex angle
is greater than 180° but less than 360°
is exactly equal to 180°
is exactly equal to 90°
is greater than 90° but less than 180°

An angle which is greater than 180° and less than 360° is called
an acute angle
an obtuse angle
a straight angle
a reflex angle

Two angles whose sum is 90° are called
supplementary angles
complementary angles
corresponding angles
alternate angles

The sum of two complementary angles is
180°
360°
90°
none of these

The angles whose sum is 180° are called
supplementary angles
complementary angles
alternate angles
corresponding angles

The sum of two supplementary angles is
90°
180°
360°
none of these

The angle complementary to 45° is
30°
45°
135°
0°

The angle complementary to 90° - 9° is
90°
90° + 9°
9°
none of these

The angle supplementary to 60° is
30°
120°
45°
300°

The angle supplementary to 90° - 9° is
90° + 9°
9°
180° - 9°
360° - 9°

The angle supplementary to 90° + 9° is
90° + 9°
90° - 9°
180° - 9°
180° + 9°

The angle supplementary to 180° - 9° is
9°
180°
180° + 9°
90° + 9°

The sum of the three angles of a triangle is
one right angle
two right angles
three right angles
four right angles

Find the angle which is equal to its complementary angle.
30°
60°
45°
90°

Find the angle which is equal to its supplementary angle.
30°
60°
45°
90°

In the following figure, the reflex angle AOB is equal to

60°
120°
300°
360°

In the following figure, ∠AOB and ∠BOC are

supplementary angles
complementary angles
adjacent angles
none of these

In the following figure, two straight lines AB and CD intersect each other at O and angles formed at O are marked. Here ∠x - ∠y has the value

56°
118°
62°
180°

In the following figure, which pair of angles is not a pair of corresponding angles?

∠1, ∠5
∠2, ∠6
∠3, ∠1
∠3, ∠5

In the following figure, a transversal n intersects two parallel lines l and m at points G and H respectively. The angles formed are marked. If ∠1 is an acute angle, tell which of the following statements is false?

∠1 + ∠2 = 180°
∠2 + ∠5 = 180°
∠3 + ∠8 = 180°
∠2 + ∠6 = 180°

In the following figure, a transversal C intersects two parallel lines a and b at points A and B. The angles formed at A and B have been marked. Tell which pair of angles need not be equal?

∠1, ∠2
∠1, ∠3
∠1, ∠5
∠2, ∠8

Which of the following is not a pair of complementary angles?

60°, 30°

56°, 34°
0°, 90°
150°, 30°

Which of the following is not a pair of supplementary angles?
90°, 90°
32°, 58°
0°, 180°
76, 104°

If m a triangle, the measures of two angles are 90° and 30°, then the measure of the third angle is
90°
30°
60°
120°

The ratio of the measures of the three angles of a triangle is 2 : 3 : 4. The measure of the largest angle is
80°
60°
40°
180°

The measure of each angle of an equilateral triangle is
90°
30°
45°
60°

The ratio of the four angles of a quadrilateral is 1 : 2 : 3 : 4. The measure of its smallest angle is
120°
36°
18°
10°

In the figure given below, the side BC of ∆ABC is extended upto D. If ∠A = 55° and ∠B = 60°, then the measure of ∠ACD is

120°
110°
115°
125°

The sum of all interior angles of a hexagon is
720°
360°
540°
1080°

The sum of all the interior angles of a pentagon is

900°
360°
540°
1080°

Which of the following can be the angles of a quadrilateral?
90°, 90°, 60°, 120°
85°, 95°, 50°, 120°
90°, 90°, 60°, 130°
90°, 90°, 50°, 140°

The measure of each angle of a regular octagon is
120°
60°
135°
108°

Which of the following are not the angles of a triangle?
45°, 45°, 90
60°, 30°, 90°
40°, 50°, 100°
60°, 60°, 60°

If two parallel lines are cut by a transversal, then which of the following is not true?
Corresponding angles are equal
Alternate interior angles are equal
Interior angles of the same side of the transversal are supplementary
Interior angles on the same side of the transversal are complementary.

One interior angle of hexagon is 165° and each of the remaining interior angles is of x°. Find the measure of each of the remaining angles.
111°
109°
107°
115°

If the measure of an angle is twice the measure of its supplementary angle then the measure of the angle is:
60°
90°
120°
130°