CBSE Class 7 Mathematics Parallel and Intersecting Lines Worksheet Set 01

Long Answer Questions

Question 1. Two straight lines AB and CD cut each other at a point O.
(i) Draw a rough figure and mark the four angles formed as ∠1, ∠2, ∠3 and ∠4, taken in order around O.
(ii) Define a linear pair and a pair of vertically opposite angles, naming one example of each from your figure.
(iii) If ∠1 = 108°, find ∠2, ∠3 and ∠4, giving the reason for each.
(iv) Show that the four angles together make 360°.
Answer:
(i) Lines AB and CD cross at O; going round O: ∠1 (upper left), ∠2 (upper right), ∠3 (lower right) and ∠4 (lower left).

(ii) A linear pair is a pair of adjacent angles whose outer arms lie along one straight line; they add up to 180°. Example: ∠1 and ∠2.
Vertically opposite angles are the two angles facing each other at the point of intersection; they are always equal. Example: ∠1 and ∠3.
(iii) ∠2 = 180° − 108° = 72°, as ∠1 and ∠2 form a linear pair.
∠3 = 108°, as ∠3 is vertically opposite ∠1.
∠4 = 72°, as ∠4 is vertically opposite ∠2.
(iv) 108° + 72° + 108° + 72° = 360°, which is one complete turn about O, so the answers are consistent.
In simple words: Two crossing lines give two angles of 108° and two of 72°, which together make 360°.

Teacher's Note:
1. Mark angles in order around the point.
2. Use one rule per angle and name it.

Teacher's Note:
1. Any sensible daily-life examples are acceptable.
2. 'Never meet' alone is not enough for parallel; same plane is needed.

Question 3. In the figure below, the transversal t cuts the lines l and m, and the eight angles formed are named a to h.
(i) What is a transversal, and how many angles does it form with a pair of lines?
(ii) Write all four pairs of corresponding angles.
(iii) Write the two pairs of alternate interior angles and the two pairs of interior angles on the same side of the transversal.
(iv) State what is true about each of these three kinds of pairs when l ∥ m.
Answer:
(i) A transversal is a line that cuts a pair of lines at two different points. It forms eight angles, four at each point.
(ii) Corresponding angles: ∠a and ∠e; ∠b and ∠f; ∠c and ∠g; ∠d and ∠h.
(iii) The interior angles (between l and m) are ∠c, ∠d, ∠e and ∠f.
Alternate interior angles: ∠c and ∠e; ∠d and ∠f.
Interior angles on the same side: ∠c and ∠f (right side); ∠d and ∠e (left side).
(iv) When l ∥ m: each pair of corresponding angles is equal; each pair of alternate interior angles is equal; each pair of same-side interior angles adds up to 180°.
In simple words: Corresponding and alternate angles are equal, and same-side interior angles add to 180°, when the lines are parallel.

Teacher's Note:
1. First find the four interior angles; the rest is easy.
2. Alternate pairs cross over the transversal; same-side pairs do not.

Question 4. Parallel and perpendicular lines can be made by folding a plain square sheet of paper.
(i) Describe how you would fold the sheet to get two creases that are parallel to each other.
(ii) Describe how you would fold the sheet to get a crease perpendicular to a crease already made.
(iii) Name one pair of parallel lines and one pair of perpendicular lines that you can see on the opened-out sheet.
(iv) Explain why the two opposite edges of the sheet and the first horizontal crease are all parallel to one another.
Answer:
(i) Fold the sheet horizontally in half and open it; one crease forms. Then bring the top edge down onto this crease, fold and open again. The two creases are parallel to each other (and to the top and bottom edges).
(ii) Fold the sheet so that one part of the existing crease falls exactly on the other part of the same crease. The new crease splits the straight angle into two equal parts of 90° each, so it is perpendicular to the first crease.
(iii) Parallel: the top edge and the first horizontal crease. Perpendicular: the first horizontal crease and the vertical crease.
(iv) The fold puts the top edge exactly on the bottom edge, so the crease is at the same distance from both edges all along. Lines that stay the same distance apart never meet, so the top edge, the crease and the bottom edge are all parallel.
In simple words: Folding edge onto edge gives parallel creases; folding a crease onto itself gives a perpendicular crease.

Teacher's Note:
1. Try the folds with a real sheet.
2. Equal distance everywhere means parallel.

Question 5. In the figure below, l ∥ m and t is a transversal. The eight angles formed are numbered as shown, and ∠6 = 145°.
(i) Find the measure of each of the eight angles.
(ii) Write the reason you used for ∠2, for ∠8 and for ∠5.
(iii) Name the two pairs of interior angles on the same side of the transversal and verify that each pair adds up to 180°.
Answer:
(i) ∠2 = ∠4 = ∠6 = ∠8 = 145°; ∠1 = ∠3 = ∠5 = ∠7 = 35°.
(ii) ∠2 = 145°: ∠2 corresponds to ∠6, and corresponding angles on parallel lines are equal.
∠8 = 145°: ∠8 is vertically opposite ∠6.
∠5 = 180° − 145° = 35°: ∠5 and ∠6 form a linear pair.
(The rest follow the same way: ∠4 is vertically opposite ∠2, ∠1 forms a linear pair with ∠2, ∠3 is vertically opposite ∠1 and ∠7 is vertically opposite ∠5.)
(iii) The interior angles are ∠3, ∠4, ∠5 and ∠6.
Right of t: ∠3 + ∠6 = 35° + 145° = 180°. Left of t: ∠4 + ∠5 = 145° + 35° = 180°.
In simple words: Four angles are 145° and the other four are 35°, and each same-side interior pair adds to 180°.

Teacher's Note:
1. One given angle decides all eight.
2. Each angle is either 145° or its supplement 35°.

Question 6. A line l is drawn on a sheet of paper and a point A is marked outside it. A line parallel to l has to be drawn through A.
(i) Write the steps for doing this with a ruler and a set square.
(ii) Write the steps for doing the same thing by paper folding.
(iii) Justify both methods using the idea of corresponding angles.
Answer:
(i) Step 1: Place one edge of the set square exactly along l.
Step 2: Hold the ruler firmly against another edge of the set square.
Step 3: Slide the set square along the ruler until the first edge reaches A.
Step 4: Draw a line along that edge through A; it is parallel to l.
(ii) Step 1: Fold to get a crease t perpendicular to l passing through A.
Step 2: Fold again to get a crease m perpendicular to t passing through A. The crease m is the required parallel line.
(iii) Set square: its slant does not change while sliding, so the angle at the old and new positions is the same. These are equal corresponding angles, so the lines are parallel.
Folding: l and m both make 90° with t. With t as transversal, the corresponding angles are both 90°, so l ∥ m.
In simple words: Both methods make equal corresponding angles, so the new line through A is parallel to l.

Teacher's Note:
1. Keep the ruler fixed; only the set square moves.
2. Two perpendiculars to the same line are parallel.

Question 7. In the figure below, AB ∥ DC and AD ∥ BC. Also, ∠DAC = 55° and ∠ADC = 70°.
(i) Find ∠DAB.
(ii) Find ∠CAB.
(iii) Find ∠BCD and ∠ABC.
(iv) Write the reason used at each step, and check that the four angles of ABCD add up to 360°.
Answer:
(i) AB ∥ DC with transversal AD, so ∠ADC and ∠DAB are same-side interior angles: 70° + ∠DAB = 180°, so ∠DAB = 110°.
(ii) Ray AC lies inside ∠DAB, so ∠DAB = ∠DAC + ∠CAB: 110° = 55° + ∠CAB, so ∠CAB = 55°.
(iii) AD ∥ BC with transversal DC, so ∠ADC + ∠BCD = 180°: ∠BCD = 110°.
AB ∥ DC with transversal BC, so ∠BCD + ∠ABC = 180°: ∠ABC = 70°.
(iv) Reasons: same-side interior angles add to 180° (parts i and iii); an angle is the sum of its parts (part ii).
Check: ∠DAB + ∠ABC + ∠BCD + ∠ADC = 110° + 70° + 110° + 70° = 360°.
In simple words: Use same-side interior angles on each pair of parallel sides to get 110°, 70°, 110° and 70°.

Teacher's Note:
1. Name the transversal each time.
2. Opposite angles of this figure come out equal.

Question 8. In the art room there are pictures in which straight lines look bent or sloping although they are actually parallel.
(i) What is meant by a parallel illusion? Describe one such picture in two or three lines.
(ii) Why does the eye get tricked in such a picture?
(iii) Describe two mathematical ways of checking whether the long lines in the picture are really parallel.
(iv) What does this tell us about trusting only the way a figure looks in geometry?
Answer:
(i) A parallel illusion is a picture in which truly parallel lines appear to slope, bend or come closer at one end. Example: a wall of black and white tiles where each row is shifted a little sideways; the long straight lines between the rows look slanted, though all of them are parallel.
(ii) Our eyes do not measure. They judge a line by comparing it with the small shapes and short slanting marks near it. When these marks tilt different ways in different places, the brain reads the long line itself as tilted.
(iii) Method 1: Measure the perpendicular distance between two long lines with a ruler at several places; equal readings mean the lines are parallel.
Method 2: Draw a straight line across them as a transversal and measure a pair of corresponding angles with a protractor; equal angles mean the lines are parallel.
(iv) In geometry, the look of a figure is never the final word. A result must be settled by careful measurement, and better still by reasoning from properties such as equal corresponding angles.
In simple words: Illusions fool the eye, so we check parallel lines by measuring gaps or corresponding angles.

Teacher's Note:
1. Name both checking methods clearly.
2. Reasoning is more reliable than looking.

Question 9. In the figure below, a water pipe runs along AB, bends at the point E, and then runs along CE up to the wall line CD. Here AB ∥ CD, ∠BAE = 35° and ∠DCE = 45°.
(i) Through E draw a line EF parallel to AB, with F lying on the left of E. Explain why EF is parallel to CD as well.
(ii) Find ∠AEF and ∠FEC, writing the reason for each.
(iii) Hence find ∠AEC.
(iv) What would ∠AEC become if both the given angles were 50°?
Answer:

(i) EF is drawn parallel to AB, and AB ∥ CD. Two lines parallel to the same line are parallel to each other, so EF ∥ CD.
(ii) AB ∥ EF with transversal AE: ∠AEF and ∠BAE are alternate interior angles, so ∠AEF = 35°.
EF ∥ CD with transversal CE: ∠FEC and ∠DCE are alternate interior angles, so ∠FEC = 45°.
(iii) Ray EF lies inside ∠AEC, so ∠AEC = 35° + 45° = 80°.
(iv) Then ∠AEF = 50° and ∠FEC = 50°, so ∠AEC = 50° + 50° = 100°. In general, the bend angle is the sum of the two given angles.
In simple words: A parallel line through the bend splits it into two alternate angles, 35° and 45°, giving 80°.

Teacher's Note:
1. Drawing EF is the key step.
2. Watch for the Z shapes that show alternate angles.

Question 10. In geometry we accept a result only after it has been reasoned out.
(i) Using corresponding angles and vertically opposite angles, explain why alternate angles formed by a transversal on a pair of parallel lines are equal.
(ii) Using the result of part (i) together with a linear pair, explain why the interior angles on the same side of the transversal add up to 180°.
(iii) A transversal cuts a pair of parallel lines and one of the interior angles measures 73°. Find the interior angle on the same side of the transversal and the alternate interior angle of the given angle.
(iv) Why is such reasoning called a proof, and why is it more dependable than measuring with a protractor?
Answer:
(The angles are named as in the figure of Long Answer Question 3, with l ∥ m.)

(i) Take the interior angle ∠f at m. Its corresponding angle at l is ∠b, so ∠f = ∠b (corresponding angles on parallel lines).
∠b and ∠d are vertically opposite, so ∠b = ∠d.
Hence ∠f = ∠b = ∠d: the alternate angles ∠f and ∠d are equal, with no measuring used.
(ii) ∠d and ∠e are the interior angles on the same (left) side. By part (i), ∠d = ∠f.
∠e and ∠f form a linear pair on m, so ∠e + ∠f = 180°. Replacing ∠f by ∠d: ∠d + ∠e = 180°.
(iii) Same-side interior angle = 180° − 73° = 107°. Alternate interior angle = 73°.
(iv) The reasoning uses only accepted properties and works for every value of the angle, not just one drawing, so it is a proof. A protractor reading can be slightly off because of line thickness or careless placing, but a proof gives a result that is exact and always true.
In simple words: Corresponding plus vertically opposite angles prove alternate angles equal, and a linear pair then gives the 180° rule.

Teacher's Note:
1. Each step must quote a known rule.
2. A proof covers all cases at once.

Case Study Based Questions

Question 11. A level crossing on a village road. The two rails of a railway track are laid on wooden sleepers so that they stay exactly the same distance apart all along the track, however far the track runs. At a level crossing near Rampur, a straight village road cuts right across both the rails. The gateman has painted the corner where the road meets the first rail, and the angle marked there, between the rail and the road, measures 65°. Nikhil stands at the gate and notices that the road makes a whole set of angles with the two rails, and that the corner at the second rail looks exactly like the corner at the first rail.
(i) What name is given to the pair of rails, and why can they never meet?
(ii) What is the village road called with respect to the two rails?
(iii) How many angles in all are formed where the road cuts the two rails?
(iv) The road makes 65° with the first rail. What is the measure of the angle in the corresponding position at the second rail? Give the reason.
Answer:
(i) The rails are a pair of parallel lines. They lie in the same plane and stay the same distance apart everywhere, so they never meet.
(ii) The road is a transversal, because it cuts both rails at two different points.
(iii) Eight angles, four at each rail.
(iv) 65°, because corresponding angles formed by a transversal on parallel lines are equal.
In simple words: The rails are parallel, the road is a transversal, and the corresponding angle is also 65°.

Teacher's Note:
1. Real-life parallel lines: rails stay the same distance apart.
2. 'Looks exactly like' hints at corresponding angles.

Question 12. Painting a zebra crossing. A road-safety team paints a zebra crossing on a straight road. The white stripes are painted so that every stripe stays the same distance from the next one, and each stripe meets the kerb line at the edge of the road at a right angle. While the painting is going on, a cycle track is also marked. The cycle track is a straight line that cuts across all the stripes, and it makes an angle of 70° with the first stripe. The team later draws the whole plan on paper, using special marks on the lines of the drawing.
(i) What is the relationship between any two white stripes?
(ii) A stripe and the kerb line meet at 90°. What are such lines called, and what are the other three angles formed there?
(iii) Which mark is used in the drawing to show that the stripes are parallel, and how is a pair of perpendicular lines marked?
(iv) What angle does the cycle track make with the fourth stripe in the corresponding position? Give the reason.
Answer:
(i) Any two stripes are parallel; they stay the same distance apart and never meet.
(ii) They are perpendicular lines. When two lines meet at 90°, all four angles are equal, so the other three angles are 90° each.
(iii) Parallel lines get a single arrow mark on each line (double arrows for a second set). Perpendicular lines are marked with a small square at the corner.
(iv) 70°, because the stripes are parallel and corresponding angles made by a transversal on parallel lines are equal.
In simple words: Stripes are parallel, each stripe is perpendicular to the kerb, and the cycle track makes 70° with every stripe.

Teacher's Note:
1. The cycle track is a transversal of all the stripes.
2. Corresponding angles stay equal across every stripe.

Question 13. A window grill. The grill of Shweta’s window is made of five long horizontal iron bars and four vertical iron bars welded to them. Each horizontal bar is the same distance from the one above it, and every vertical bar meets each horizontal bar at a right angle. For decoration, the blacksmith has welded one more straight bar which runs slantwise across all five horizontal bars. The slanting bar makes an angle of 55° with the topmost horizontal bar, on the right side and below it.
(i) What is the relationship between any two horizontal bars of the grill?
(ii) What is the relationship between a horizontal bar and a vertical bar, and what is the measure of the angle between them?
(iii) What is the slanting bar called with respect to the horizontal bars?
(iv) At the second horizontal bar, what is the measure of the alternate interior angle of the given 55° angle, and what is the measure of the angle that forms a linear pair with it?
Answer:
(i) Any two horizontal bars are parallel.
(ii) A horizontal bar and a vertical bar are perpendicular; the angle between them is 90°.
(iii) The slanting bar is a transversal of the horizontal bars.
(iv) The alternate interior angle is 55° (alternate angles on parallel lines are equal). The angle forming a linear pair with it is 180° − 55° = 125°.
In simple words: The bars form parallel and perpendicular lines; the alternate angle is 55° and its linear pair is 125°.

Teacher's Note:
1. Below the top bar and above the second bar is the interior region.
2. Linear pair: subtract from 180°.

Question 14. A folding activity with a newspaper. In the mathematics period the class is given a square sheet cut from a newspaper. The teacher asks the children to look carefully at the four edges of the sheet first, and then to fold the sheet horizontally into half, open it out, and fold it horizontally into half once more. Each fold leaves a straight crease on the sheet. The children then count the straight lines they can see running from the left edge to the right edge, including the two original horizontal edges. Finally the teacher asks them to make one vertical fold in the same sheet.
(i) How are the opposite edges of the square sheet related to each other?
(ii) How are two adjacent edges of the sheet related, and what is the angle between them?
(iii) After the two horizontal folds, how many parallel lines can be counted on the sheet?
(iv) The new vertical crease is ______ to the horizontal creases and ______ to the vertical edges of the sheet.
Answer:
(i) Opposite edges are parallel.
(ii) Adjacent edges are perpendicular; the angle between them is 90°.
(iii) 5 parallel lines: the two horizontal edges and three creases.
(iv) The new vertical crease is perpendicular to the horizontal creases and parallel to the vertical edges of the sheet.
In simple words: Opposite edges are parallel, adjacent edges are perpendicular, and two folds give 5 parallel lines.

Teacher's Note:
1. The second fold adds two creases, not one.
2. A vertical crease runs the same way as the side edges.

Question 15. A street plan of a colony. The plan of a new colony is drawn on a chart. MG Road and Nehru Road are laid straight and are always the same distance apart, so they never meet. Station Road is laid straight too, and it cuts MG Road at the junction P and Nehru Road at the junction Q. At P, the angle between Station Road and MG Road, marked above the road on the right side, measures 110°. Two angles at Q have been marked x° and y° on the plan, as shown.
(i) Which road acts as the transversal, and which two roads are parallel?
(ii) Find the value of x, giving the reason.
(iii) Find the value of y, giving the reason.
(iv) A new road is to be laid perpendicular to MG Road. Will it also be perpendicular to Nehru Road? Give the reason.
Answer:
(i) Station Road is the transversal. MG Road and Nehru Road are parallel, as shown by the arrow marks.
(ii) x = 110. The 110° angle at P and x° at Q are in the same position (above the road, right of Station Road), so they are corresponding angles, which are equal on parallel lines.
(iii) y = 70. x° and y° are adjacent angles along Nehru Road, so they form a linear pair: y = 180 − 110 = 70.
(iv) Yes. The new road makes 90° with MG Road. Taking it as a transversal of the two parallel roads, the corresponding angle at Nehru Road is also 90°, so it is perpendicular to Nehru Road too.
In simple words: x equals the corresponding 110°, y is its linear pair 70°, and a road perpendicular to one parallel road is perpendicular to the other.

Teacher's Note:
1. Find x first, then use it for y.
2. A perpendicular to one of two parallel lines is perpendicular to both.

Question 16. A ladder against a wall. A painter rests a ladder against the wall of a house. The ladder has two long straight side rails which are kept exactly the same distance apart all along their length, and a number of rungs are fixed across them. Each rung is fixed at a right angle to both rails. When the ladder is set in position, the first rail makes an angle of 62° with the level ground on the side away from the wall. The ground itself is a straight level line that cuts across both rails.
(i) What is the relationship between the two side rails, and between any two rungs?
(ii) What is the relationship between a rung and a side rail?
(iii) The ground acts as a transversal of the two rails. What angle does the second rail make with the ground in the corresponding position, and why?
(iv) Find the angle that the first rail makes with the ground on the other side of its foot.
Answer:
(i) The two side rails are parallel, and any two rungs are also parallel.
(ii) A rung is perpendicular to a side rail.
(iii) 62°, because the ground is a transversal of the parallel rails and corresponding angles are equal.
(iv) 180° − 62° = 118°, because the two angles at the foot form a linear pair along the ground.
In simple words: The rails are parallel, so the second rail also makes 62°, and the angle on the other side is 118°.

Teacher's Note:
1. Rungs are perpendicular to the rails, so they are parallel to each other.
2. Angles on a straight line add to 180°.

Question 17. A wooden staircase. A carpenter builds a wooden staircase. The flat surfaces on which we step, called treads, are all made level, so each tread is parallel to the others. The upright boards between two treads, called risers, are fixed at right angles to the treads. A long straight handrail runs over the whole flight of steps. When the lines of the treads are extended, the handrail cuts across them and makes an angle of 32° with the first tread, measured on the right side above the tread.
(i) Name the relationship between two treads, and between a tread and a riser.
(ii) The handrail is a transversal of the treads. What angle does it make with the next tread in the corresponding position?
(iii) Find the interior angle on the same side of the handrail at the next tread.
(iv) On another staircase the carpenter measures 32° at one tread and 34° at the next, both in the corresponding position. What does this tell him about those two treads?
Answer:
(i) Two treads are parallel; a tread and a riser are perpendicular.
(ii) 32°, because the treads are parallel and corresponding angles are equal.
(iii) The same-side interior angle of the given 32° angle, at the next tread, is 180° − 32° = 148°, since same-side interior angles add up to 180°.
(iv) The corresponding angles 32° and 34° are not equal, so those two treads are not parallel; one of them is not level and must be corrected.
In simple words: The handrail makes 32° with every level tread; unequal angles show a tread is not level.

Teacher's Note:
1. Equal corresponding angles test for parallel treads.
2. Same-side interior angles are supplementary.

Question 18. A ruled notebook page. Meera has a notebook with ruled lines printed across every page. The lines are printed by a machine, so they are always the same distance apart and never come closer together. A red margin line is printed down the page, and it meets every ruled line at a right angle. For a design, Meera draws a straight pencil line slanting across four ruled lines. The pencil line makes an angle of 54° with the first ruled line, measured on the right side below that line.
(i) What is the relationship between the ruled lines, and between the margin line and a ruled line?
(ii) The pencil line is a transversal of the ruled lines. What angle does it make with the fourth ruled line in the corresponding position?
(iii) At the second ruled line, which angle is the alternate interior angle of the given 54° angle, and what is its measure?
(iv) Meera says that the red margin line is also a transversal of the ruled lines, and that every angle it makes with them is a right angle. Is she correct? Explain.
Answer:
(i) The ruled lines are parallel; the margin line is perpendicular to each ruled line.
(ii) 54°, because the ruled lines are parallel and corresponding angles are equal.
(iii) It is the angle above the second ruled line, on the left side of the pencil line, and it measures 54°.
(iv) Yes. The margin line cuts all the ruled lines, so it is a transversal. It meets the first ruled line at 90°, and since the ruled lines are parallel, every corresponding angle is also 90°; all the angles are right angles.
In simple words: The pencil line makes 54° with every ruled line, and the margin line makes right angles with all of them.

Teacher's Note:
1. Alternate angles switch both side and position (below-right becomes above-left).
2. A perpendicular transversal makes all eight angles 90°.

Question 19. An illusion on the art-room wall. A poster on the art-room wall shows row after row of small black and white squares. In each row the squares are shifted a little to the right of the row below it. Between the rows run long straight lines. Every child in the class says that these long lines slope, some going up and some going down, and that they would certainly meet somewhere. The art teacher smiles and says that all the long lines on the poster are parallel to one another, and that the eye is being fooled by the shifted squares. She then asks the class to settle the matter using the geometry they have learnt.
(i) What are such pictures called?
(ii) How can a ruler alone be used to check whether the long lines are really parallel?
(iii) How can a straight line drawn across them, together with a protractor, settle the question? Which property would you use?
(iv) Why does the poster trick our eyes, and what does this teach us about geometry?
Answer:
(i) They are called parallel illusions (optical illusions about parallel lines).
(ii) Measure the perpendicular distance between two long lines at several places. If every reading is the same, the lines are parallel.
(iii) Draw a straight line across all the long lines as a transversal and measure a pair of corresponding angles at two long lines. If they are equal, the lines are parallel. Property used: equal corresponding angles mean the lines are parallel.
(iv) The shifted squares put short edges pointing different ways near each long line, and the eye judges the long line by these edges, so it looks sloping. Geometry teaches us not to trust looks alone; we must measure carefully and, better still, reason from properties.
In simple words: The lines only look sloping; equal gaps or equal corresponding angles prove they are parallel.

Teacher's Note:
1. Two tests: constant distance and equal corresponding angles.
2. Illusions show why proofs matter.

Question 20. A roof truss. A roof truss is made of two long straight beams AB and CD which are fixed exactly the same distance apart, so they never meet. Two slanting braces are welded between them. The first brace runs from the point P on AB down to the point Q on CD, and the angle ∠BPQ measures 58°. The second brace runs from the point R on AB down to the point S on CD, and the angle ∠BRS measures 122°. The engineer must now decide whether the two braces are parallel to each other.
(i) Name the two transversals in the figure and state what the arrow marks on AB and CD mean.
(ii) Find ∠PQC, giving the reason.
(iii) Find ∠PQD and show that the interior angles on the same side of the brace PQ add up to 180°.
(iv) Taking AB as a transversal of the two braces, decide whether PQ is parallel to RS. Give the reason.
Answer:
(i) The braces PQ and RS are the transversals, as each cuts both beams. The arrow marks show that AB ∥ CD.
(ii) ∠PQC = 58°. With PQ as transversal of the parallel beams, ∠BPQ and ∠PQC are alternate interior angles, which are equal.
(iii) ∠PQC and ∠PQD form a linear pair on CD, so ∠PQD = 180° − 58° = 122°.
The same-side interior angles of brace PQ are ∠BPQ and ∠PQD: 58° + 122° = 180°.
(iv) With AB as transversal of PQ and RS, ∠BPQ = 58° and ∠BRS = 122° are in the same position at P and R, so they are corresponding angles. They are not equal, so PQ is not parallel to RS; extended, the braces would meet.
In simple words: Alternate angles give ∠PQC = 58°, and unequal corresponding angles 58° and 122° show the braces are not parallel.

Teacher's Note:
1. The same line can be a transversal for different pairs.
2. Here AB is the transversal for the two braces.