Chapter 6: Lines and Angles

(C) 145°

Explanation:

According to the given figure, we have

AB || CD || EF

PQ || RS

∠RQD = 25°

∠CQP = 60°

PQ || RS.

We know that,

If a transversal intersects two parallel lines, then each pair of alternate exterior angles is equal.

Now, since PQ || RS

⇒ ∠PQC = ∠BRS

We have ∠PQC = 60°

⇒ ∠BRS = 60° … eq.(i)

We also know that,

If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.

Now again, since AB || CD

⇒ ∠DQR = ∠QRA

We have ∠DQR = 25°

⇒ ∠QRA = 25° … eq.(ii)

Using linear pair axiom,

We get,

∠ARS + ∠BRS = 180°

⇒ ∠ARS = 180° – ∠BRS

⇒ ∠ARS = 180° – 60° (From (i), ∠BRS = 60°)

⇒ ∠ARS = 120° … eq.(iii)

Now, ∠QRS = ∠QRA + ∠ARS

From equations (ii) and (iii), we have,

∠QRA = 25° and ∠ARS = 120°

Hence, the above equation can be written as:

∠QRS = 25° + 120°

⇒ ∠QRS = 145°

Therefore, option (C) is the correct answer.

(D) A right triangle

Explanation:

Let the angles of △ABC be ∠A, ∠B and ∠C

Given that ∠A= ∠B+∠C …(eq1)

But, in any △ABC,

Using the angle sum property, we have,

∠A+∠B+∠C=180^o …(eq2)

From equations (eq1) and (eq2), we get

∠A+∠A=180^o

⇒2∠A=180^o

⇒∠A=180^o/2 = 90^o

⇒∠A = 90^o

Hence, we get that the triangle is a right triangle

Therefore, option (D) is the correct answer.

(B) 52 ½^o

Explanation:

According to the question,

The exterior angle of triangle = 105°

Let the two interior opposite angles of the triangle = x

We know that,

The exterior angle of a triangle = sum of interior opposite angles

Then, we have the equation,

105° = x + x

2x = 105°

x = 52.5°

x = 52½

Therefore, option (B) is the correct answer.

(A) An acute angled triangle

Explanation:

According to the question,

The angles of a triangle are of the ratio 5 : 3 : 7

Let 5:3:7 be 5x, 3x and 7x

Using the angle sum property of a triangle,

5x + 3x +7x =180

15x=180

x=12

Substituting the value of x, x = 12, in 5x, 3x and 7x we get,

5x = 5×12 = 60^o

3x = 3×12 = 36^o

7x = 7×12 = 84^o

Since all the angles are less than 90^o, the triangle is an acute-angled triangle.

Therefore, option (A) is the correct answer.