Chapter 5: Parallel and Intersecting Lines (Geometry)

No. Two different straight lines can meet at most at one point (if they met at two points, they would be the same line).

For lines meeting at angles $a,b,c,d$ (in order around the point):

• Opposite angles are equal: $\angle a=\angle c$ and $\angle b=\angle d$.
• Neighbouring angles add up to 180°: $\angle a+\angle b=\angle b+\angle c=\angle c+\angle d=\angle d+\angle a=180°$.
• All four together make 360°.

Yes, always: $\angle a+\angle b=180°$ and $\angle a+\angle d=180°$ (straight angles), so $\angle b=\angle d$; similarly $\angle a=\angle c$, whatever the size of $\angle a$.

No. Vertically opposite angles are equal and adjacent angles (linear pairs) add up to 180°, for every pair of intersecting lines.

Linear pairs: (a, b), (b, c), (c, d), (d, a). Vertically opposite: (b, d), (a, c).

Yes. The four angles together make 360°, so each is $360°÷4=90°$. Such lines are perpendicular.

Yes; each angle is 360° ÷ 4 = 90° (perpendicular lines).

• FG and FH meet at their common endpoint F at an angle of 115.3°.
• AB and CD intersect at the point X, which is not an endpoint of either; X lies near the middle of AB but is not its midpoint. The angles formed measure about 60° and 120° (opposite angles equal).
• IJ and LM intersect at the point Y (not at an endpoint), forming angles of about 80° and 100°.
• ST and UV: they come slightly closer together towards S and U. If extended far enough to the left, they will meet.
• OP and QR run in exactly the same direction and stay the same distance apart. They will not meet: they are parallel.

(Measure the angles in your book with a protractor; small differences from these values are expected.)

FG, FH meet at endpoint F (115.3°); AB, CD intersect at X; IJ, LM intersect at Y; ST and UV would meet if extended (to the left); OP and QR are parallel and will not meet.

In the classroom: opposite edges of the blackboard, door, window frames and desk tops; the lines on a ruled notebook page; the edges of floor tiles; window grill bars; the steps of a staircase.

Fig. 5.6: a, i and h (all vertical); c and g (horizontal); d and f (slanting the same way down); b and e (slanting the same way up).

Classroom: board edges, notebook lines, tile edges, grills. Fig. 5.6: a, i, h; c and g; d and f; b and e.

• Opposite edges are parallel; adjacent edges are perpendicular (they meet at right angles).
• After 1 horizontal fold: 3 parallel lines (two edges + the fold), and the fold is perpendicular to the vertical sides.
• After 2 folds: 5; after 3 folds: 9; after 4 folds: 17. Each fold doubles the number of strips (2, 4, 8, 16), and the number of lines is one more than the number of strips: $2^n+1$ after $n$ folds.
• A vertical fold is perpendicular to all the horizontal lines.
• Yes: fold one of the two corners that are not on the diagonal so that it touches the centre of the square. The crease is parallel to the diagonal joining the other two corners.

Opposite edges parallel, adjacent perpendicular; 3, 5, 9, 17 lines (2ⁿ + 1); a vertical fold is perpendicular to them; yes, a fold bringing a corner to the centre is parallel to a diagonal.

Yes.

• a and p lie along the left and right edges of the square, which are parallel.
• b and q lie along the two folds made towards the centre line; both are parallel to the edges (and to the centre crease), so b ∥ q.
• c and r are the slanting edges of the two folded corner triangles; the two corners are folded in exactly the same way (at the same angle) from opposite sides, so c and r make equal angles with the vertical creases and are parallel.

Yes: a, p are opposite edges; b, q are folds parallel to the centre line; c, r are made by identical corner folds, so they make equal angles with the creases.

• A horizontal segment: draw a vertical one through the dots.
• A vertical segment: draw a horizontal one.
• A segment going 1 dot right and 1 dot up each step (45°): draw one going 1 right and 1 down.
• A segment going 1 right and 2 down: draw one going 2 right and 1 up. (In general, for "a right, b up", the perpendicular goes "b right, a down".)

Turn each direction through 90°: horizontal ↔ vertical; "1 right, 1 up" ↔ "1 right, 1 down"; "1 right, 2 down" ↔ "2 right, 1 up".

(a) Perpendicular lines: where a horizontal side of a shape meets a vertical side (along the grid lines), they make a right angle; also two slanting sides where one goes "1 right, 1 up" and the other "1 right, 1 down".

(b) Parallel lines: sides that go the same number of squares across and up (the same "steps") are parallel, e.g. the two slanting sides of the parallelogram, opposite sides of the rectangles, and all horizontal sides with each other.

(a) A horizontal side meeting a vertical side, or 45° slants in opposite directions. (b) Sides with the same steps across and up (same direction) are parallel.

Draw each set with the same "step" from one endpoint to the other:

• Set 1: horizontal segments in different rows.
• Set 2: segments each going 1 dot right and 1 dot up.
• Set 3: segments each going 2 right and 1 down (any length that is a whole number of such steps).

Use the same "across and up" step for all lines in a set, e.g. all 1 right–1 up, all 2 right–1 down.

(a) Yes, for some of them.

(b) The horizontal, vertical and 45° segments (a, b, c, d) were easy. The ones at unusual slopes, e, f, g and h, were harder.

(c) Count the steps of the given segment from one end to the other (for example, "1 right, 3 down" or "4 right, 1 up") and repeat the same steps from a new starting dot. Then the new segment has the same direction and is parallel.

(a) Yes (b) e, f, g, h (c) By copying the same "across and down" steps from another dot.

Line c. Counting the steps along each line (or measuring the angle each makes with the line joining their left endpoints to the bottom dot), c rises and runs in the same ratio as a and makes the same angle, while b is less steep and would meet a if both were extended to the left. Equal corresponding angles with a transversal show that a ∥ c.

c; it has the same slope as a (equal corresponding angles), while b is flatter.

No. The eight angles form four pairs of vertically opposite angles (∠1 = ∠3, ∠2 = ∠4, ∠5 = ∠7, ∠6 = ∠8), so there are at most four different measures.

Five different values for ∠6, ∠5, ∠4, ∠3, ∠2 is also impossible, since ∠2 = ∠4.

No: vertically opposite angles are equal, so there are at most 4 different angles; ∠2 = ∠4 rules out five.

Both lines make 90° with l. Taking l as a transversal, the corresponding angles are equal (90° each), so the two lines are parallel.

With the long side of the set square, both new lines make the same angle (e.g. 30° or 60°) with the ruler, because the set square was only slid along the ruler. Tracing or measuring shows the corresponding angles are equal, so the lines are parallel.

The corresponding angles they make with l (or with the ruler) are equal, so the lines are parallel.

Method (ruler and set square):

1. Place one edge of the set square exactly along line l.
2. Hold a ruler firmly against another edge of the set square.
3. Slide the set square along the ruler (without turning it) until its first edge reaches point A.
4. Draw a line through A along that edge. This line is parallel to l, because it makes the same (corresponding) angle with the ruler as l does.

(Another way: draw any line through A meeting l; measure the angle it makes with l, and copy the same angle at A in the corresponding position.)

Slide a set square along a fixed ruler from l to A and draw along its edge; equal corresponding angles make the line parallel to l.

Both l and m are perpendicular to t. With t as a transversal, the corresponding angles at the two intersections are both 90°, i.e. equal. So l ∥ m.

Both are perpendicular to t, so the corresponding angles are 90° = 90°, and l ∥ m.

$\angle 3+\angle 6=50°+130°=180°$.

Proof: ∠2 and ∠3 are a linear pair, so $\angle 2=180°-\angle 3$. Since $l\parallel m$, the corresponding angles ∠2 and ∠6 are equal, so $\angle 6=180°-\angle 3$, i.e. $\angle 3+\angle 6=180°$ for any value of ∠3.

The interior angles on the same side of the transversal always add up to 180°.

The arrow-marked lines are parallel in each figure.

a = 48°, b = 52°, c = 81°, d = 99°, e = 69°, f = 48°, g = 122°, h = 75°, i = 54°, j = 97°

(i) The angle a and the 42° angle are on the same line in the "above-left / above-right" positions of corresponding angles: $a=180°-42°=$ 138°. (The 100° is at the other transversal.)

(ii) The transversals are parallel too. The 62° and a sit at the same type of position on opposite sides: $a=180°-62°=$ 118°.

(iii) The slanting line makes 110° with the top parallel, i.e. 70° on the other side. Its angle with the other line at the middle parallel is 35°, so the second line makes $110°-35°=75°$ with the parallels on that side, and $a=180°-75°=$ 105°.

(iv) The two arrowed lines are parallel, so the right one also makes 67° with the base. In the right-angled triangle formed with the vertical line: $a=180°-90°-67°=$ 23°.

(i) 138° (ii) 118° (iii) 105° (iv) 23°

(i) The vertical line is perpendicular to the parallels, so the slanting line makes $90°-65°=25°$ with the lower parallel. x is vertically opposite to this angle: x = 25°. At the upper parallel, y is the angle on the other side of the slanting line: y = 180° − 25° = 155°.

(ii) The two crossing lines make 53° and 78° with the parallel lines (corresponding angles at the upper line). The angle between them above the crossing point is $x=78°-53°=$ 25°.

(i) x = 25°, y = 155° (ii) x = 25°

The lines IA and GD are parallel; CH and JF are transversals.

• ∠GEH is vertically opposite to ∠DEB, which corresponds to ∠ABC: ∠GEH = 45°.
• ∠FED is vertically opposite to ∠KEG, which corresponds to ∠IKJ: ∠FED = 78°.
• G, E, D lie on a straight line: $\angle HEF=180°-45°-78°=$ 57°.

∠GEH = 45°, ∠HEF = 57°, ∠FED = 78°

With BE as the transversal of the parallel lines CD and EF, the angle at D between DC (upwards) and DE is the alternate angle of ∠BEF $=55°$. y is on the other side of the line at D: $y=180°-55°=$ 125°.

x at B is the corresponding angle of y (AB ∥ CD): x = 125°.

x = y = 125°

Draw lines through N and O parallel to LM (and PQ).

• At N: the part of ∠MNO on the M side equals the alternate angle ∠LMN $=40°$, so the other part is $96°-40°=56°$.
• At O: the part of ∠NOP on the N side is the alternate angle of this 56°, i.e. $56°$; the part on the P side is the alternate angle of ∠OPQ, i.e. $52°$.

$\angle NOP=56°+52°=$ 108°

∠NOP = 108°

There are parallel lines: the two vertical lines in the first picture, the long slanting stripes in the second and the two horizontal lines in the third are all parallel (check with a ruler).

The illusion is caused by the background lines: many lines crossing or radiating from a point make the parallel lines appear to bend towards or away from each other. Our brain judges a line's direction by comparing it with the lines around it, so the surrounding pattern tricks our eyes. Measuring the distance between the lines at two places shows it is the same.

Yes, they are parallel (check with a ruler); the crossing or radiating background lines make our eyes see them as bent or slanting.