Mathematics Solutions Solutions for Class 8 Math Chapter 1 Rational And Irrational Numbers are provided here with simple step-by-step explanations. These solutions for Rational And Irrational Numbers are extremely popular among Class 8 students for Math Rational And Irrational Numbers Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Mathematics Solutions Book of Class 8 Math Chapter 1 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Mathematics Solutions Solutions. All Mathematics Solutions Solutions for class Class 8 Math are prepared by experts and are 100% accurate.

#### Page No 8:

#### Question 1:

In the adjoining figure, each angle is shown by a letter. Fill in the boxes with the help of the figure.

Corresponding angles.

(1) ∠p and ☐

(2) ∠q and ☐

(3) ∠r and ☐

(4) ∠s and ☐

Interior alternate angles.

(5) ∠s and ☐

(6) ∠w and ☐

#### Answer:

Corresponding angles : If the arms on the transversal of a pair of angles are in the same direction and the other arms are on the same side of transversal, then it is called a pair of corresponding angles.

Corresponding angles

(1) ∠*p* and $\overline{)\angle w}$

(2) ∠*q* and $\overline{)\angle x}$

(3) ∠*r *and $\overline{)\angle y}$

(4) ∠*s* and $\overline{)\angle z}$

Alternate interior angles : A pair of angles which are on the opposite side of the transversal and inside the given lines that are intersected by the transversal.

Interior alternate angles

(5) ∠*s* and $\overline{)\angle x}$

(6) ∠*w *and $\overline{)\angle r}$

#### Page No 8:

#### Question 2:

Observe the angles shown in the figure and write the following pair of angles.

(1) Interior alternate angles

(2) Corresponding angles

(3) Interior angles

#### Answer:

(1) Alternate interior angles : A pair of angles which are on the opposite side of the transversal and inside the given lines that are intersected by the transversal.

Interior alternate angles

(a) ∠*c* and ∠*e *

(b) ∠*b* and ∠h

(2) Corresponding angles : If the arms on the transversal of a pair of angles are in the same direction and the other arms are on the same side of transversal, then it is called a pair of corresponding angles.

Corresponding angles

(a) ∠*d* and ∠*h *

(b) ∠*c* and ∠*g*

(c) ∠*a* and ∠*e*

(d) ∠*b* and ∠*f*

(3) Interior angles : A pair of angles which are on the same side of transversal and inside the given lines that are intersected by the transversal.

Interior angles

(a) ∠*c* and ∠*h*

(b) ∠*b* and ∠*e*

#### Page No 11:

#### Question 1:

1. Choose the correct alternative.

(1) In the adjoining figure, if line *m* ∥ line *n* and line *p* is a transversal then find *x*.

(A) 135$\xb0$

(B) 90$\xb0$

(C) 45$\xb0$

(D) 40$\xb0$

(2) In the adjoining figure, if line *a *∥ line *b* and line *l *is a transversal then find *x*.

(A) 90$\xb0$

(B) 60$\xb0$

(C) 45$\xb0$

(D) 30$\xb0$

#### Answer:

(1)

Let us mark the points P and Q on *m*; R and S on *n*; A and B on *p*.

Suppose PQ and AB intersect at M; RS and AB intersect at N.

Since, *m*||*n *and *p* is a transversal, then

*m*∠QMN + *m*∠SNM = 180° (Interior angles on the same side of transversal are supplementary)

Substituing the values in the above equation, we get

3*x* + *x* = 180°

⇒ 4*x* = 180°

⇒ *x* = $\frac{180\xb0}{4}$

∴ *x* = 45°

So, the correct answer is option (C).

(2)

Let us mark the points P and Q on *a*; R and S on *b*; A and B on *l*.

Suppose PQ and AB intersect at M; RS and AB intersect at N.

Since *a*||*b* and *l* is a transversal, then

*m*∠RNM = *m*∠SNB (Vertically opposite angles)

⇒ ∠RNM = 2*x*

Now, *m*∠RNM + *m*∠PMN = 180° (Interior angles on the same side of transversal are supplementary)

⇒ 2*x* + 4*x *= 180°

⇒ 6*x* = 180°

⇒ *x* = $\frac{180\xb0}{6}$

⇒ *x* = 30°

So, the correct answer is option (D).

#### Page No 11:

#### Question 2:

In the adjoining figure line *p *∥ line *q*. Line *t* and line *s* are transversals. Find measure of ∠*x* and ∠*y* using the measures of angles given in the figure.

#### Answer:

Let us mark the points P and Q on *p*; R and S on *q*; A and B on *t*; C and D on *s*.

Suppose PQ and AB intersect at K; PQ and CD intersect at X.

Suppose RS and AB intersect at L; RS and CD intersect at Y.

Since, AB is a straight line and ray KQ stands on it,

*m*∠AKX + *m*∠XKL = 180° (angles in linear pair)

⇒ 40° + *m*∠XKL = 180°

⇒ *m*∠XKL = 180° − 40°

⇒ *m*∠XKL = 140°

Since, *p*||*q* and *t* is a transversal, then

*m*∠YLB = *m*∠XKL (Corresponding angles)

⇒ *x* = 140°

Since, RS and CD are two straight lines intersecting at Y, then

*m*∠XYL = *m*∠SYD (Vertically opposite angles)

⇒ *m*∠XYL = 70°

Since, *p*||*q* and *s* is a transversal, then

*m*∠KXY + *m*∠XYL = 180° (Interior angles on same side of transversal are supplementary)

⇒ *y* + 70° = 180°

⇒ *y* = 180° − 70°

⇒ *y* = 110°

#### Page No 12:

#### Question 3:

In the adjoining figure. line *p* ∥ line *q*. line *l* ∥ line *m*. Find measures of ∠*a*, ∠*b* and ∠*c*, using the measures of given angles. Justify your answers.

#### Answer:

Let us mark the points A and B on *p*; X and Y on *q*; P and Q on *l*; R and S on *m*.

Suppose AB and XY intersect PQ at K and L respectively.

Suppose AB and XY intersect RS at N and M respectively.

Since, *p*||*q* and *l* is a transversal, then

*m*∠AKL + *m*∠XLK = 180° (Interior angles on same side of transversal are supplementary)

⇒ 80° + *m*∠XLK = 180°

⇒ *m*∠XLK = 180° − 80°

⇒ *m*∠XLK = 100°

Since, PQ and XY are straight lines that intersect at L, then

*m*∠QLM = *m*∠XLK (Vertically opposite angles)

⇒ *a* = 100°

Since, *l*||*m* and *p* is a transversal, then

*m*∠BNR = *m*∠AKL (Alternate exterior angles)

⇒ *c* = 80°

Since, *p*||*q* and *m* is a transversal, then

*m*∠NMY= *m*∠RNB (Corresponding angles)

⇒ *b* = *c*

⇒ *b* = 80°

#### Page No 12:

#### Question 4:

In the adjoining figure, line *a* ∥ line *b*. Line *l* is a transversal. Find the measures of ∠*x*, ∠*y*, ∠*z* using the given information.

#### Answer:

Let us mark the points A and B on *l*; K and M on *a*; L and N on *b*.

Suppose KM and LN intersect AB at P and Q respectively.

Since, *a*||*b* and *l* is a transversal, then

*m*∠PQL = *m*∠APK (Corresponding angles)

⇒ *x* = 105°

Since, AB and LN are straight lines that intersect at Q, then

*m*∠BQN = *m*∠PQL (Vertically opposite angles)

⇒ *y* = *x*

⇒ *y* = 105°

Since, AB is a straight line and ray QN stands on it, then

*m*∠BQN + *m*∠PQN = 180° (Angles in linear pair)

⇒ y + *m*∠PQN = 180°

⇒ 105° + *m*∠PQN = 180°

⇒ *m*∠PQN = 180° − 105°

⇒ *m*∠PQN = 75°

Now,* m*∠APM = *m*∠PQN (Corresponding angles)

⇒ *z* = 75°

#### Page No 12:

#### Question 5:

In the adjoining figure, line *p* ∥ line *l* ∥ line *q*. Find ∠*x* with the help of the measures given in the figure.

#### Answer:

Let us mark the points A, L and B on *p*; C, M and D on *l*; P, N and Q on *q*.

Since, AB||CD and LM is a transversal intersecting AB at L and CD at M, then

*m*∠LMD = *m*∠ALM (Alternate interior angles)

⇒ *m*∠LMD = 40°

Since, CD||PQ and MN is a transversal intersecting CD at M and PQ at N, then

*m*∠DMN = *m*∠PNM (Alternate interior angles)

⇒ *m*∠DMN = 30°

Now, *m*∠LMD + *m*∠DMN = 40° + 30°

⇒ *m*∠LMN = 70°

⇒ *x* = 70°

#### Page No 13:

#### Question 1:

Draw a line *l*. Take a point A outside the line. Through point A draw a line parallel to line *l*.

#### Answer:

Steps of construction :

(1) Draw a line *l*. Take a point A outside the line *l*.

(2) Draw a segment AM ⊥ line *l*.

(3) Take another point N on line *l*.

(4) Draw a segment NB ⊥ line *l*, such that *l*(NB) = *l*(MA).

(5) Draw a line *m* passing through the points A and B.

Hence, the line *m* is the required line that passes through point A and parallel to line *l*.

#### Page No 13:

#### Question 2:

Draw a line *l*. Take a point T outside the line. Through point T draw a line parallel to line *l*.

#### Answer:

Steps of construction :

(1) Draw a line *l*. Take a point T outside the line *l*.

(2) Draw a segment MT ⊥ line *l*.

(3) Take another point N on line *l*.

(4) Draw a segment NV ⊥ line *l*, such that *l*(NV) = *l*(MT).

(5) Draw a line *m* passing through the points T and V.

Hence, the line *m* is the required line that passes through point T and parallel to line *l*.

#### Page No 13:

#### Question 3:

Draw a line *m*. Draw a line *n *which is parallel to line *m *at a distance of 4 cm from it.

#### Answer:

Steps of construction :

(1) Draw a line *m*.

(2) Take two points A and B on the line *m*.

(3) Draw perpendiculars to the line *m* at A and B.

(4) On the perpendicular lines, take points P and Q at a distance of 4 cm from A and B respectively.

(5) Draw a line *n* passing through the points P and Q.

So, line *n* is the required line parallel to the line *m* at a distance of 4 cm away from it.

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