Parallel and Intersecting Lines · Lesson 2 of 6
Recognising Parallel Lines and Folding Patterns
“Recognise, draw, mark and create parallel and perpendicular line families on paper.”
• Describe segment meetings using endpoints, intersections and midpoints. • Explain parallel lines and the same-plane condition. • Use arrow marks and right-angle squares correctly. • Investigate parallel and perpendicular folds and explain the fold-count pattern. • Draw parallel and perpendicular segments in different directions on dot paper.
Segments, Endpoints and Extensions
The lines we study have no ends, but the lines we draw occupy a finite sheet. That is why two drawn segments might not meet even though their extended straight lines do. Before calling them parallel, imagine extending both ends and consider the directions they follow.
An endpoint is one end of a segment. Its midpoint is the point halfway between its endpoints, dividing it into two equal lengths. An intersection point is not automatically a midpoint.
When three letters name an angle, the middle letter is its vertex. In ∠GFH, the angle is at F between FG and FH. In the source arrangement, those segments meet at their common endpoint F and the angle measures 115.3°. Two other segments, AB and CD, cross at X; IJ and ML cross at Y. Say “intersect at X” or “intersect at Y”. Say “midpoint” only after equal lengths have been given or checked.
Problem
Two short segments do not touch. Is that enough information to say that their lines are parallel?
- 1.No. A segment has endpoints, whereas its supporting line extends in both directions.
- 2.Use a ruler to extend the segments. If their directions lead to a common point, the supporting lines intersect, even if the point lies outside the original drawing.
- 3.Only if their supporting lines lie in the same plane and never meet can we call those lines parallel.
Compare the segment pairs ST/UV and OP/QR in the next diagram. This recreates the extension investigation: the dashed parts continue the straight directions beyond the endpoints. ST and UV meet only after extension, while the supporting lines of OP and QR are marked parallel. Both pairs look separated as short segments, but that observation alone does not decide parallelism.
Extend a drawn pair carefully with a ruler, keeping each extension straight. If their directions differ, their supporting lines will meet somewhere, perhaps beyond the drawing. If their distinct supporting lines are parallel, they will never meet. Later, angle relationships will give a precise test without requiring extremely long extensions.
What Parallel Means
Look at the straight edges of a keyboard, the slats of a bench and the rails of a fence in the same flat face. Many such edges run in the same direction. On one plane, two distinct lines that never meet, however far extended, are parallel.
Distinct lines in the same plane that do not meet when extended without limit in either direction. The symbol ∥ means “is parallel to”. Parallel segments are segments whose supporting lines are parallel.
Parallelism is about direction and nonintersection, not segment length. A short horizontal segment can be parallel to a much longer horizontal one. The perpendicular distance between two parallel lines stays constant. Measure the gap perpendicular to the lines; an arbitrary slanted distance between selected points is not the distance between the lines.
A line on a tabletop and a line on a different board might never meet. That fact alone does not make them parallel: the definition also requires both lines to lie in one plane. In the flat paper constructions here, the same-plane condition is already satisfied.
Find examples on one classroom board or sheet: opposite edges of a rectangular notice, lines of writing, or straight grid lines. Parallel strokes also make shading in art. Check which edges belong to the same flat face rather than comparing every visible line in a room.
Problem
On dot paper, a segment moves 2 columns right and 1 row up. A second moves 6 columns right and 3 rows up. Can they be parallel?
- 1.The second movement repeats the first movement three times: 6 right and 3 up has the same direction as 2 right and 1 up.
- 2.Place the segments on different supporting lines of that direction. Their lengths differ, but their lines are parallel.
- 3.If they were placed on the same supporting line, they would be collinear rather than a pair of distinct parallel lines. Position matters as well as direction.
Create Parallel Lines by Repeated Folding
A square has two pairs of opposite parallel edges. Adjacent edges are perpendicular because they meet at 90°. Repeatedly halving the folded sheet horizontally makes parallel creases, and unfolding reveals a number pattern that is easy to miscount if we forget the edges.
Use a square sheet. Fold it horizontally in half, then keep the sheet folded and halve it horizontally again. Repeat once more and then unfold completely. Count horizontal creases separately from the two horizontal edges. Next make a vertical fold: its crease is perpendicular to every horizontal crease. Finally fold along a diagonal and try to create a second crease of the same diagonal direction. Compare the directions, not their positions on the sheet.
| Successive halvings | Equal horizontal strips | Interior horizontal creases | Parallel lines including top and bottom edges |
|---|---|---|---|
| 1 | 2 | 1 | 3 |
| 2 | 4 | 3 | 5 |
| 3 | 8 | 7 | 9 |
| 4 | 16 | 15 | 17 |
Each new halving doubles the number of strips in the unfolded sheet: 2, 4, 8, 16. A row of 8 strips has 7 boundaries inside it, plus its top and bottom boundaries. Thus after three halvings there are 7 interior creases and 9 horizontal lines including the two edges. The next count is 17, not 18: the pattern doubles the strips, rather than doubling every boundary.
If n is the number of successive horizontal halvings, 2ⁿ means multiplying 2 by itself n times. There are 2ⁿ equal strips, so there is one fewer interior crease than strips. Adding the two outside edges gives one more line than strips.
Problem
After three successive horizontal halvings, what changes after a fourth?
- 1.There are 8 strips after three halvings. A fourth halving doubles that to 16 strips.
- 2.Sixteen strips have 15 interior crease lines. Including the two horizontal edges gives 15 + 2 = 17 parallel lines.
- 3.The counts refer to the fully unfolded sheet. Counting the folded packet itself would hide most creases.
A Second Folding Investigation
Parallel lines can also appear when corners are folded in matching ways. Keep a square sheet on the table so that all its edges and creases are on the same plane. We will compare pairs of directions in two triangular corner folds.
Fold the square along its vertical centre and unfold. Bring each vertical side to the centre crease, fold, then unfold to leave quarter-width guides. Fold the top-right corner diagonally inward to make a right triangular flap in the rightmost strip, and the bottom-left corner similarly in the leftmost strip. The flaps should not cross their inner strip guides. Label the upper vertical, horizontal and diagonal flap edges a, b, c; label the matching lower edges p, q, r. Compare a with p, b with q, and c with r. Try unfolding again to inspect each crease.
Edges a and p follow parallel vertical directions. Edges b and q are both horizontal, each perpendicular to the vertical guides, so they are parallel. The two matching triangular folds have diagonals in the same direction, making c and r parallel too. This conclusion depends on making the matching folds as described; arbitrary folds of two corners need not have parallel diagonal edges.
Marks That Communicate Geometry
A picture can be approximate, so we need symbols to say which relationships are given. Matching arrow marks on two lines mean those lines are parallel. Use a different number of arrow marks for a second parallel family; a small square at an intersection means 90°.
On dot paper, horizontal and vertical directions are easiest to compare. For a slant, count the horizontal and vertical moves between endpoints and repeat the same move from another dot. For example, “3 right, 2 up” can be copied anywhere on the grid. A 45° direction uses equal horizontal and vertical moves. Other slants may be harder to reproduce by eye; counting grid moves or sliding a set square is more dependable.
Problem
A given segment runs 2 dots right and 1 dot up. Draw a perpendicular direction.
- 1.Use 1 dot right and 2 dots down for the second direction. It is a quarter-turn of the first grid movement.
- 2.Draw its line through a chosen dot and extend until it crosses the original supporting line. Check the right angle using the square corner of a set square.
- 3.This illustrates a grid construction. Horizontal/vertical and the two opposite 45° slants are simpler special cases of the same quarter-turn idea.
In a source comparison of three nearby slanted lines a, b and c, c is parallel to a. Line b has a slightly different direction. Visual recognition suggests the choice; repeating grid movements, measuring perpendicular gaps or using the angle test in the next lesson helps verify it. Do not choose the line merely because it is nearer.
Check Your Understanding
Choose an answer using the geometry, then explain your choice. The explanations after the questions show the relationship that justifies each answer.
Quiz
Two segments have different lengths. When can they be parallel?
After three successive horizontal halvings, how many horizontal interior creases appear when unfolded?
What does a small square at an intersection state?
Which description of a midpoint is correct?
Two nonmeeting lines on different flat surfaces must be parallel. Is this valid?
Which grid move repeats the direction 3 right, 2 up?
Segment length does not determine parallelism. The supporting lines must be distinct, coplanar and nonmeeting.
Try these questions on paper. Label your diagram, record the given information, and name the reason for each angle calculation or construction. Compare your reasoning with the solutions after your first attempt.
Practice Problems
- Draw perpendiculars to a horizontal segment, a vertical segment and a 45° slanted segment on dot paper.
- In the dot-paper diagram, add parallel arrows to a/b, c/d and e/f. Mark a right angle at the extended intersection of a and c. Explain how you recognised them.
- Draw three pairs of parallel segments with dots as endpoints. Make the two segments of one pair different lengths.
- Draw a parallel to each of these eight directions: horizontal, vertical, 45° up-right, 45° down-right, 2 right/1 up, 1 right/2 up, 3 right/1 down, 1 right/3 down. Which were harder?
- Three lines a, b and c are drawn so that a and c have the same direction, while b is slightly steeper. Which is parallel to a, and how could you check?
- Predict creases and horizontal lines including edges after five successive halvings.
- In the triangular-fold investigation, explain all three matching pairs without relying on segment length.
- Describe FG and FH meeting at F with ∠GFH = 115.3°. Is F an endpoint or necessarily a midpoint?
Use a vertical line for the horizontal segment, a horizontal line for the vertical one, and the opposite 45° slant for the third. Extend the lines to cross and check each right angle with a set square.
Key Takeaways
• Parallel lines are distinct, coplanar and never meet even when extended. • Segments of different lengths can have parallel supporting lines. • An intersection is a midpoint only when it divides a segment into equal halves. • Matching arrows mark parallel families; a square marks a right angle. • Successive horizontal halvings produce 2ⁿ − 1 creases and 2ⁿ + 1 horizontal lines including edges. • Matching grid moves or controlled folds preserve direction; appearance alone needs checking.