NCERT Exemplar solutions for Mathematics [English] Class 9 chapter 6 - Lines & Angles [Latest edition]

In the following figure, if AB || CD || EF, PQ || RS, ∠RQD = 25° and ∠CQP = 60°, then ∠QRS is equal to ______.

If one angle of a triangle is equal to the sum of the other two angles, then the triangle is ______.

An exterior angle of a triangle is 105° and its two interior opposite angles are equal. Each of these equal angles is ______.

The angles of a triangle are in the ratio 5 : 3 : 7. The triangle is ______.

If one of the angles of a triangle is 130°, then the angle between the bisectors of the other two angles can be ______.

In the following figure, POQ is a line. The value of x is ______.

In the following figure, if OP || RS, ∠OPQ = 110° and ∠QRS = 130°, then ∠PQR is equal to ______.

Angles of a triangle are in the ratio 2 : 4 : 3. The smallest angle of the triangle is ______.

For what value of x + y in figure will ABC be a line? Justify your answer.

Can a triangle have all angles less than 60°? Give reason for your answer.

Can a triangle have two obtuse angles? Give reason for your answer.

How many triangles can be drawn having its angles as 45°, 64° and 72°? Give reason for your answer.

How many triangles can be drawn having its angles as 53°, 64° and 63°? Give reason for your answer.

In the following figure, find the value of x for which the lines l and m are parallel.

Two adjacent angles are equal. Is it necessary that each of these angles will be a right angle? Justify your answer.

If one of the angles formed by two intersecting lines is a right angle, what can you say about the other three angles? Give reason for your answer.

In the following figure, which of the two lines are parallel and why? 

Two lines l and m are perpendicular to the same line n. Are l and m perpendicular to each other? Give reason for your answer.

In the following figure, OD is the bisector of ∠AOC, OE is the bisector of ∠BOC and OD ⊥ OE. Show that the points A, O and B are collinear.

In the following figure, ∠1 = 60° and ∠6 = 120°. Show that the lines m and n are parallel.

AP and BQ are the bisectors of the two alternate interior angles formed by the intersection of a transversal t with parallel lines l and m (Figure). Show that AP || BQ.

In the following figure, bisectors AP and BQ of the alternate interior angles are parallel, then show that l ||m.

In the following figure, BA || ED and BC || EF. Show that ∠ABC = ∠DEF

[Hint: Produce DE to intersect BC at P (say)].

In the following figure, BA || ED and BC || EF. Show that ∠ABC + ∠DEF = 180°

In the following figure, DE || QR and AP and BP are bisectors of ∠EAB and ∠RBA, respectively. Find ∠APB.

The angles of a triangle are in the ratio 2 : 3 : 4. Find the angles of the triangle.

A triangle ABC is right angled at A. L is a point on BC such that AL ⊥ BC. Prove that ∠BAL = ∠ACB.

Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.

If two lines intersect, prove that the vertically opposite angles are equal.

Bisectors of interior ∠B and exterior ∠ACD of a ∆ABC intersect at the point T. Prove that `∠BTC = 1/2 ∠BAC`.

A transversal intersects two parallel lines. Prove that the bisectors of any pair of corresponding angles so formed are parallel.

Prove that through a given point, we can draw only one perpendicular to a given line.

[Hint: Use proof by contradiction].

Prove that two lines that are respectively perpendicular to two intersecting lines intersect each other.

[Hint: Use proof by contradiction].

Prove that a triangle must have atleast two acute angles.

In the following figure, ∠Q > ∠R, PA is the bisector of ∠QPR and PM ⊥ QR. Prove that ∠APM = `1/2` (∠Q – ∠R).