# Class 9 RD Sharma Solutions – Chapter 8 Introduction to Lines and Angles- Exercise 8.4 | Set 1

**Question 1: In figure, AB, CD and âˆ 1 and âˆ 2 are in the ratio 3 : 2. Determine all angles from 1 to 8.**

**Solution:**

Assume,

âˆ 1 = 3x and âˆ 2 = 2x

From the following figure: âˆ 1 and âˆ 2 are a linear pair of angles

Thus,

âˆ 1 + âˆ 2 = 180

Â°3x + 2x = 180

Â°5x = 180

Â°x =

x = 36

Â°Hence,

âˆ 1 = 3x = 108

Â°and

âˆ 2 = 2x = 72

Â°We know that:

(Vertically opposite angles are equal)

Here the pairs of vertically opposite angles are:

(âˆ 1 = âˆ 3);

(âˆ 2 = âˆ 4);

(âˆ 5, âˆ 7)

and

(âˆ 6 = âˆ 8)

âˆ 1 = âˆ 3 = 108Â°

âˆ 2 = âˆ 4 = 72Â°

âˆ 5 = âˆ 7

âˆ 6 = âˆ 8

Now as it is known

(if a transversal intersects any parallel lines, then the corresponding angles are equal)

âˆ 1 = âˆ 5 = âˆ 7 = 108Â°

âˆ 2 = âˆ 6 = âˆ 8 = 72Â°

Hence,

âˆ 1 = 108Â°, âˆ 2 = 72Â°, âˆ 3 = 108Â°, âˆ 4 = 72Â°, âˆ 5 = 108Â°, âˆ 6 = 72Â°, âˆ 7 = 108Â° and âˆ 8 = 72Â°

**Question 2: In figure, I, m**,** and n are parallel lines intersected by transversal p at X, Y and Z respectively. Find âˆ 1, âˆ 2**,** and âˆ 3.**

**Solution:**

Here as given in the figure

âˆ Y = 120Â° {Vertical opposite angles]}

âˆ 3 + âˆ Y = 180Â° {Linear pair angles}

âˆ 3= 180 â€“ 120

â‡’ âˆ 3= 60Â°

As we can see the line ‘l’ is parallel to line ‘m’,

âˆ 1 = âˆ 3 {Corresponding angles}

âˆ 1 = 60Â°

Now, line ‘m’ is parallel to line ‘n’,

âˆ 2 = âˆ Y {Alternate interior angles}

âˆ 2 = 120Â°

âˆ 1 = 60Â°,

âˆ 2 = 120Â°

and

âˆ 3 = 60Â°.

**Question 3: In figure, AB || CD || EF and GH || KL. Find âˆ HKL.**

**Solution:**

Construct: Extend LK to meet line GF at point P.

As shown below.

Here as from the figure,

CD || GF,

âˆ CHG =âˆ HGP = 60Â° {alternate angles}

âˆ HGP =âˆ KPF = 60Â° {Corresponding angles of parallel lines}

Thus,

âˆ KPG =180

Â°â€“ 60Â°â‡’âˆ KPG = 120

Â°âˆ GPK = âˆ AKL= 120Â° {Corresponding angles of parallel lines}

âˆ AKH = âˆ KHD = 25Â° {alternate angles of parallel lines}

Thus,

âˆ HKL = âˆ AKH + âˆ AKL

â‡’25 + 120

â‡’âˆ HKL = 145Â°

**Question 4: In figure, show that AB || EF.**

**Solution:**

Construct: Produce EF to intersect AC at point N.

As it is seen in the figure:

âˆ BAC = 57Â°

and

âˆ ACD = 22Â°+35Â° = 57Â°

{Alternative angles of parallel lines are equal}

BA || EF â€¦..(i)

We know that,

Sum of Co-interior angles of parallel lines is 180Â°

EF || CD

âˆ DCE + âˆ CEF = 35 + 145 = 180Â° â€¦(ii)

From (i) and (ii)

AB || EF {Lines parallel to the same line are parallel to each other}

Hence Proved.

**Question 5: In figure, if AB || CD and CD || EF, find âˆ ACE.**

**Solution:**

Given:

CD || EF

âˆ FEC + âˆ ECD = 180Â° {Sum of co-interior angles is supplementary to each other}

âˆ ECD = 180Â° â€“ 130Â° = 50Â°

Now, BA || CD

âˆ BAC = âˆ ACD = 70Â° {Alternative angles of parallel lines are equal}

Thus,

âˆ ACE + âˆ ECD =70Â°

âˆ ACE = 70Â° – 50Â°

âˆ ACE = 20

Â°

**Question 6: In figure, PQ || AB and PR || BC. If âˆ QPR = 102Â°, determine âˆ ABC. Give reasons.**

**Solution:**

Construct:

Extend line AB to meet line PR at point G.

As shown below;

Given:

PQ || AB,

âˆ QPR = âˆ BGR =102Â° {Corresponding angles of parallel lines}

And

PR || BC,

âˆ RGB+ âˆ CBG =180Â° {Corresponding angles are supplementary}

âˆ CBG = 180Â° â€“ 102Â° = 78Â°

Thus,

âˆ CBG = âˆ ABC

â‡’ âˆ ABC = 78Â°

**Question 7: In figure, state which lines are parallel and why?**

**Solution:**

As we know that,

If a transversal intersects two lines such that a pair of alternate interior angles are equal, then the two lines are parallel

As we can see from the figure:

â‡’ âˆ EDC = âˆ DCA = 100Â°

Lines DE and AC are intersected by a transversal DC such that the pair of alternate angles are equal.

Hence,

DE || AC

**Question 8: In figure, if l||m, n || p and âˆ 1 = 85Â°, find âˆ 2.**

**Solution:**

Given:

âˆ 1 = 85Â°

We know that,

When a line cuts the parallel lines, the pair of alternate interior angles are equal.

â‡’ âˆ 1 = âˆ 3 = 85Â°

Thus again, co-interior angles are supplementary,

Therefore

âˆ 2 + âˆ 3 = 180Â°

âˆ 2 + 55Â° =180Â°

âˆ 2 = 180Â° â€“ 85Â°

âˆ 2 = 95Â°

**Question 9: If two straight lines are perpendicular to the same line, prove that they are parallel to each other.**

**Solution:**

Assume lines ‘l’ and ‘m’ are perpendicular to ‘n’,

Thus

âˆ 1= âˆ 2=90Â°

Therefore,

The lines ‘l’ and ‘m’ are cut by a transversal line i.e. ‘n’

The corresponding angles are equal, so it can be seen that,

Line ‘l’ is parallel to line ‘m’.

**Question 10: Prove that if the two arms of an angle are perpendicular to the two arms of another angle, then the angles are either equal or supplementary.**

**Solution:**

Consider the angles be âˆ ACB and âˆ ABD

Let AC perpendicular to AB, and

CD is perpendicular to BD.

To Prove:

âˆ ACD = âˆ ABD

âˆ ACD + âˆ ABD =180Â°

Proof :

In a quadrilateral,

âˆ A+ âˆ C+ âˆ D+ âˆ B = 360Â° {Sum of angles of quadrilateral is 360Â°}

180Â° + âˆ C + âˆ B = 360Â°

âˆ C + âˆ B = 360Â° â€“180Â°

Thus,

âˆ ACD + âˆ ABD = 180Â°

And

âˆ ABD = âˆ ACD = 90Â°

Therefore, angles are equal as well as supplementary.