Parallel & Intersecting Lines

Can two straight lines intersect at more than one point?

Answer

No.

Two straight lines can intersect at only one point. If they intersected at more than one point, they would coincide and become the same line.

Hence, two distinct straight lines cannot intersect at more than one point.

What patterns do you observe among the angles formed by two intersecting lines?

Answer

The two intersecting lines l and m are shown below:

The vertically opposite angles are equal:

∠a = ∠c

∠b = ∠d

Also, each pair of adjacent angles forms a linear pair and adds up to 180°:

∠a + ∠b = 180°

∠b + ∠c = 180°

∠c + ∠d = 180°

∠d + ∠a = 180°

Hence, vertically opposite angles are equal, while each linear pair adds up to 180°.

In Fig. 5.2, if ∠a is 120°, can you figure out the measurements of ∠b, ∠c and ∠d without drawing and measuring them?

Answer

Given:

∠a = 120°

Since ∠a and ∠b form a linear pair,

∠a + ∠b = 180°

120° + ∠b = 180°

⇒ ∠b = 180° − 120°

⇒ ∠b = 60°

Since ∠b and ∠c form a linear pair,

∠b + ∠c = 180°

60° + ∠c = 180°

⇒ ∠c = 180° − 60°

⇒ ∠c = 120°

Since ∠c and ∠d form a linear pair,

∠c + ∠d = 180°

120° + ∠d = 180°

⇒ ∠d = 180° − 120°

⇒ ∠d = 60°

Hence, ∠b = 60°, ∠c = 120°, and ∠d = 60°.

Is it always true that the vertically opposite angles formed by a pair of intersecting lines are equal?

Answer

Yes. We can reason this without assuming any particular value of ∠a.

Since a straight angle measures 180°, the linear pairs give:

∠a + ∠b = 180° and ∠a + ∠d = 180°

∴ ∠b = ∠d

Similarly:

∠b + ∠a = 180° and ∠b + ∠c = 180°

∴ ∠a = ∠c

So the vertically opposite angles are equal for every measure of ∠a.

Hence, for any pair of intersecting lines, the vertically opposite angles are equal.

List all the linear pairs and vertically opposite angles you observe in Fig. 5.3:

Answer

When the two lines intersect, four angles ∠a, ∠b, ∠c and ∠d are formed. Adjacent angles form linear pairs, and opposite angles are vertically opposite.

Hence, there are four linear pairs and two pairs of vertically opposite angles.

Can you draw a pair of intersecting lines such that all four angles are equal? Can you figure out what will be the measure of each angle?

Answer

Yes, we can draw a pair of intersecting lines such that all four angles are equal.

The four angles formed at a point of intersection together make one complete turn, so they add up to 360°.

If all four angles are equal, then each angle is:

⇒ $\dfrac{360^\circ}{4} = 90^\circ$

Thus, each angle is a right angle, and the two lines are perpendicular to each other.

Hence, Yes, a pair of intersecting lines can be drawn such that all four angles are equal, and each angle measures 90°.

Observe Fig. 5.5 and describe the way the line segments meet or cross each other in each case, with appropriate mathematical words (a point, an endpoint, the midpoint, meet, intersect) and the degree measure of each angle.

(i) Are line segments ST and UV likely to meet if they are extended?
(ii) Are line segments OP and QR likely to meet if they are extended?

Answer

We describe each pair using the given example as a model.

For example, line segments FG and FH meet at the endpoint F at an angle of 115.3°.

In the same way:

• Line segments AB and CD intersect at point X.
• Line segments IJ and LM intersect at point Y.
• Line segments ST and UV do not meet in the figure.
• Line segments OP and QR do not meet in the figure.

(i) Extending line segments ST and UV, we see that they are not parallel — they slope towards each other and would meet on the right-hand side.

∴ Yes, line segments ST and UV are likely to meet if they are extended.

(ii) Line segments OP and QR stay almost the same distance apart, so they are parallel.

∴ No, line segments OP and QR are not likely to meet even if they are extended.

Hence, ST and UV are likely to meet on extension, while OP and QR appear parallel and are not likely to meet.

Which pairs of lines appear to be parallel in Fig. 5.6 below?

Answer

Looking at the directions of the drawn lines, the lines that never seem to meet (they stay the same distance apart) are grouped as follows:

• Lines a, i and h appear to be parallel to one another.

• Lines c and g appear to be parallel.

• Lines d and f appear to be parallel.

• Lines e and b appear to be parallel.

Hence, the parallel lines are: a, i and h; c and g; d and f; and e and b.

Draw some lines perpendicular to the lines given on the dot paper in Fig. 5.10.

Answer

Lines perpendicular to the given lines on the dot paper are drawn as shown below:

In Fig. 5.11, mark the parallel lines using the notation given above (single arrow, double arrow etc.). Mark the angle between perpendicular lines with a square symbol.
(a) How did you spot the perpendicular lines?
(b) How did you spot the parallel lines?

Answer

On Fig. 5.11, each set of parallel lines is marked with the same arrow notation — a single arrow (>) on the first set of parallel lines, a double arrow (>>) on the second set, and so on. Every pair of perpendicular lines is marked with a small square symbol placed at the right angle between them.

(a) How did you spot the perpendicular lines?

The vertical and horizontal lines of the grid meet each other at a 90° angle, i.e. a right angle. A pair of lines that intersect to form a right angle are perpendicular to each other.

Hence, the perpendicular lines were spotted by the right angle (90°) formed between them.

(b) How did you spot the parallel lines?

The parallel lines were identified as the lines that always remain the same distance apart and never meet, however far they are extended.

Hence, the parallel lines were spotted as the lines that stay the same distance apart throughout.

In the dot paper following, draw different sets of parallel lines. The line segments can be of different lengths but should have dots as endpoints.

Answer

Within a set, each line must run in exactly the same direction as the others, so that the lines stay the same distance apart and never meet when extended. The line segments in a set may have different lengths, as long as their endpoints fall on the dots.

Using your sense of how parallel lines look, try to draw lines parallel to the line segments on this dot paper.
(a) Did you find it challenging to draw some of them?
(b) Which ones?
(c) How did you do it?

Answer

(a) Yes, some of them were challenging to draw.

(b) The slanted line segments e, f, g and h were more difficult to draw parallel to accurately.

(c) Each parallel line was drawn by keeping it equidistant from the given line segment along its whole length, so that the two lines never come closer together or move farther apart.

In Fig. 5.13, which line is parallel to line a — line b or line c? How do you decide this?

Answer

Line b slopes towards line a, so if both are extended they would eventually meet — they are not parallel.

Line c, however, stays the same distance from line a everywhere along its length and runs in exactly the same direction as line a. So the two lines will never meet, no matter how far they are extended.

Hence, line c is parallel to line a.

Is it possible for all the eight angles to have different measurements? Why, why not?

(i) What about five different angles — 6, 5, 4, 3 and 2?

Answer

When a transversal t crosses two lines, eight angles are formed — four at each intersection.

At each intersection, the vertically opposite angles are equal to each other:

∠1 = ∠3, ∠2 = ∠4, ∠5 = ∠7, and ∠6 = ∠8

Because of these equalities, the eight angles can have a maximum of four distinct measures. So they cannot all have different measurements.

Hence, no — it is not possible for all eight angles to be different, since vertically opposite angles are equal.

(i) Among the angles ∠2, ∠3, ∠4, ∠5 and ∠6, the angles ∠2 and ∠4 are vertically opposite angles, so they are equal:

∠2 = ∠4

Since two of these five angles are always equal, they cannot all have five different measurements.

Hence, no — five different angle measures are not possible, because ∠2 = ∠4.

How do you know these two lines are parallel? Can you check if the corresponding angles are equal?

Answer

The two lines are drawn using the same edge of the set square. Therefore, each line makes the same angle with line l.

If line l is considered as a transversal, then the angles formed with the two lines are corresponding angles.

Since these corresponding angles are equal, the two lines are parallel.

We can verify this by tracing the angles or by measuring them with a protractor. In both cases, the corresponding angles are found to be equal.

Hence, the two lines are parallel because the corresponding angles formed with the transversal are equal.

Can you draw a line parallel to l, that goes through point A? How will you do it with the tools from your geometry box? Describe your method.

Answer

Yes. We can draw the required line using a ruler and a set square.

Tools required: Ruler, set square (right-angled triangle), pencil and eraser.

Steps:

1. Place one edge of the set square along the line l.

2. Hold the ruler firmly against another edge of the set square so that the ruler does not move.

3. Slide the set square along the ruler until the first edge reaches point A.

4. Draw a line along this edge of the set square, through point A.

5. This new line passes through A and is parallel to line l.

Why are lines l and m parallel to each other?

Answer

While folding the paper, we first make the crease t perpendicular to l (passing through A). Then we make the crease m perpendicular to t (passing through A).

So both l and m are perpendicular to the same line t.

Now treat t as a transversal cutting l and m. Each of these lines makes an angle of 90° with t, so the corresponding angles formed are equal.

∴ Since the corresponding angles are equal, l and m are parallel.

Hence, lines l and m are parallel because both are perpendicular to the common line t, which makes their corresponding angles equal (90° each).

In Fig. 5.25, if ∠f is 120° what is the measure of its alternate angle ∠d?

Answer

Given:

∠f = 120°

The alternate angle of ∠f is found by first taking the corresponding angle of ∠f, and then taking the vertically opposite angle of that.

∠b = ∠f = 120° $\quad$[∠b is the corresponding angle of ∠f]

∴ ∠d = ∠b = 120° $\quad$[∠d is the vertically opposite angle of ∠b]

Hence, the measure of ∠d is 120°.

Is there a relation between ∠3 and ∠6? Try to find the relationship by taking different values for ∠3 and see what ∠6 is. Once you find a relation, try to justify it or prove that this relation holds always.

Answer

∠3 and ∠6 are the interior angles lying on the same side of the transversal, formed where it crosses the two parallel lines.

On taking different values of ∠3 and finding the matching ∠6 each time, we notice that their sum is always 180°.

This can be justified. ∠2 and ∠3 form a linear pair, so:

∠2 = 180° − ∠3

Also, ∠2 and ∠6 are corresponding angles between the parallel lines, so:

∠6 = ∠2 = 180° − ∠3

∴ ∠3 + ∠6 = ∠3 + (180° − ∠3) = 180°

Hence, the interior angles on the same side of the transversal, ∠3 and ∠6, always add up to 180°.

Find the angle marked below.

Answer

The marked angle a and the 48° angle are alternate interior angles formed by a transversal intersecting parallel lines.

∴ a = 48°

Find the angle marked below.

Answer

The marked angle b and the 52° angle are alternate angles.

∴ b = 52°

Find the angle marked below.

Answer

The marked angle c and the 81° angle are alternate angles.

∴ c = 81°

Find the angle marked below.

Answer

The marked angle d and the 99° angle are alternate angles.

∴ d = 99°

Find the angle marked below.

Answer

The marked angle e and the 69° angle are alternate angles.

∴ e = 69°

Find the angle marked below.

Answer

The marked angle f and the 132° angle are interior angles on the same side of the transversal.

∴ f + 132° = 180°

⇒ f = 180° − 132°

⇒ f = 48°

Find the angle marked below.

Answer

The marked angle g and the 122° angle are corresponding angles.

∴ g = 122°

Find the angle marked below.

Answer

The marked angle h and 75° angle are alternate angles.

∴ h = 75°

Find the angle marked below.

Answer

The marked angle i and 54° angle are alternate angles.

∴ i = 54°

Find the angle marked below.

Answer

The marked angle j and the 97° are alternate interior angles formed by a transversal intersecting parallel lines.

∴ j = 97°

Find the angle represented by a.

Answer

In the figure,

∠CDH = ∠EDF $\quad$ (vertically opposite angles)

∴ ∠CDH = 42°

The marked angle a and ∠CDH are interior angles on the same side of the transversal.

∴ a + 42° = 180°

⇒ a = 180° − 42°

⇒ a = 138°

Find the angle represented by a.

Answer

In the figure,

∠FEG and ∠EGH are interior angles on the same side of the transversal.

∴ ∠FEG + ∠EGH = 180°

⇒ 62° + ∠EGH = 180°

⇒ ∠EGH = 180° - 62°

⇒ ∠EGH = 118°

The marked angle a and ∠EGH are exterior alternate angles.

∴ a = ∠EGH

⇒ a = 118°

Find the angle represented by a.

Answer

In the figure,

Given: ∠EGB = 35°

∠CBE + ∠EBG = 180° $\quad$ (Linear pair)

⇒ 110° + ∠EBG = 180°

⇒ ∠EBG = 180° - 110°

⇒ ∠EBG = 70°

Now, ∠BGM = ∠EBG $\quad$ (Interior alternate angles)

∴ ∠BGM = 70°

The marked angle a and ∠EGM are corresponding angles.

∴ a = ∠EGM

⇒ a = ∠EGB + ∠BGM

⇒ a = 35° + 70°

⇒ a = 105°

Find the angle represented by a.

Answer

In the figure,

Given:

∠BFA = 67° and ∠CFE = 90°

Since AFE is a straight line,

∠BFA + ∠BFC + ∠CFE = 180°

67° + ∠BFC + 90° = 180°

157° + ∠BFC = 180°

∠BFC = 180° - 157°

∠BFC = 23°

Now, ∠BFE = ∠CDE $\quad$ (corresponding angles)

∠BFE = ∠BFC + ∠CFE

∠BFE = 23° + 90°

∠BFE = 113°

∴ ∠CDE = 113°

∠CDE + ∠CDF = 180° $\quad$ (straight angle)

113° + ∠CDF = 180°

∠CDF = 180° - 113°

∠CDF = 67°

In △CFD,

∠CFD + ∠CDF + ∠DCF = 180°

⇒ 90° + 67° + a = 180°

⇒ 157° + a = 180°

⇒ a = 180° - 157°

⇒ a = 23°

In the figures below, what angles do x and y stand for?

Answer

(i)

In the figure,

Given:

∠BHD = 65°, ∠CBD = 90°, ∠KHJ = x and ∠FDH = y

∠CBD = ∠BHG = 90° $\quad$ (corresponding angles)

∠BHD + ∠DHG = 90°

⇒ 65° + ∠DHG = 90°

⇒ ∠DHG = 90° - 65°

⇒ ∠DHG = 25°

∠KHJ = ∠DHG $\quad$ (vertically opposite angles)

∴ x = 25°

∠DHG + ∠FDH = 180° $\quad$ (interior angles on the same side of the transversal)

25° + y = 180°

⇒ y = 180° - 25°

⇒ y = 155°

Hence, x = 25° and y = 155°

(ii)

In the figure,

Given:

∠BGF = 78°, ∠BJG = 53° and ∠CBD = x

∠BGF + ∠BGJ = 180°

78° + ∠BGJ = 180°

∠BGJ = 180° - 78°

∠BGJ = 102°

In △ BJG,

∠BJG + ∠BGJ + ∠GBJ = 180°

⇒ 53° + 102° + ∠GBJ = 180°

⇒ 155° + ∠GBJ = 180°

⇒ ∠GBJ = 180° - 155°

⇒ ∠GBJ = 25°

∠CBD = ∠GBJ $\quad$ (vertically opposite angles)

∴ x = 25°

In Fig. 5.33, ∠ABC = 45° and ∠IKJ = 78°. Find angles ∠GEH, ∠HEF, ∠FED.

Answer

Here IA is parallel to GD, and the two transversals through the points K and B meet the line GD at the same point E.

For ∠GEH:

The line C–B–E–H is a transversal of the parallel lines IA and GD. ∠GEH and ∠ABC are alternate angles.

∴ ∠GEH = ∠ABC = 45°

For ∠FED:

The line J–K–E–F is a transversal of the parallel lines IA and GD. ∠FED and ∠IKJ are alternate angles.

∴ ∠FED = ∠IKJ = 78°

For ∠HEF:

G, E and D lie on a straight line, so ∠GEH, ∠HEF and ∠FED together make 180°.

∠GEH + ∠HEF + ∠FED = 180°

⇒ 45° + ∠HEF + 78° = 180°

⇒ 123° + ∠HEF = 180°

⇒ ∠HEF = 180° − 123°

⇒ ∠HEF = 57°

Hence, ∠GEH = 45°, ∠HEF = 57° and ∠FED = 78°.

In Fig. 5.34, AB is parallel to CD and CD is parallel to EF. Also, EA is perpendicular to AB. If ∠BEF = 55°, find the values of x and y.

Answer

The lines AB, CD and EF are all parallel, and EB is a transversal cutting across them.

x and ∠BEF are interior angles on the same side of the transversal EB.

∴ x + ∠BEF = 180°

⇒ x + 55° = 180°

⇒ x = 180° − 55°

⇒ x = 125°

Since CD is parallel to AB, x and y are corresponding angles.

∴ y = x = 125°

Hence, x = 125° and y = 125°.

What is the measure of angle ∠NOP in Fig. 5.35?
[Hint: Draw lines parallel to LM and PQ through points N and O.]

Answer

Following the hint, draw a line through N parallel to LM and a line through O parallel to PQ. These auxiliary lines let us split ∠NOP into two parts, each of which can be matched to a known angle using alternate angles.

In the figure,

Given:

∠QPO = 52°, ∠LMN = 40°, ∠MNO = 96° and ∠NOP = a

∠QPO = ∠POT = 52° $\quad$ (alternate angles)

∠LMN = ∠MNS = 40° $\quad$ (alternate angles)

∠SNO = ∠MNO - ∠MNS

⇒ ∠SNO = 96° - 40°

⇒ ∠SNO = 56°

∠NOT = ∠SNO $\quad$ (alternate angles)

∴ ∠NOT = 56°

Here,

∠NOP = ∠NOT + ∠POT

⇒ ∠NOP = 56° + 52°

⇒ ∠NOP = 108°

Hence, the measure of ∠NOP is 108°.