Parallel and Intersecting Lines · Lesson 3 of 6
Transversals, Corresponding Angles, and Parallel Constructions
“Use matching angle positions to test parallelism and explain reliable constructions.”
• Identify a transversal and the eight angles at its two crossings. • Explain why eight angles have at most four distinct measures. • Identify corresponding angles by relative position. • Use both directions of the corresponding-angle and parallel-line relationship. • Construct a parallel through a given point with tools or paper folding and justify the result.
One Line Crosses Two Others
A single crossing gives four angles. Now draw a line that crosses two different lines at different points: there are two crossings, and hence two sets of four angles. The crossing line connects the two sets and lets us compare their positions.
A line that intersects two or more lines at distinct points. The two lines it crosses need not be parallel. In this lesson a transversal t crosses l and m at two different points.
Use the diagram to name positions consistently. At the upper crossing, 1 is above l on the left of t, 2 above l on the right, 3 below l on the right, and 4 below l on the left. At the lower crossing, 5, 6, 7 and 8 occupy the matching positions around m. If you rotate the page, rotate this whole arrangement; do not relabel an angle according to what happens to look “up” afterward.
At the first crossing, 1 = 3 and 2 = 4 because vertically opposite angles are equal. At the second, 5 = 7 and 6 = 8. Each crossing therefore contributes at most two different measures. Together the eight angles can have at most four distinct measures. They cannot all be different, and they cannot have five different measures. Even the proposed set 6, 5, 4, 3 and 2 repeats a measure because 2 = 4.
Problem
A transversal crosses two lines that are not assumed parallel. One crossing has an angle of 60° and the other an angle of 80°. List the possible measures.
- 1.At the first crossing, the opposite is 60° and each neighbour is 180° − 60° = 120°.
- 2.At the second crossing, the opposite is 80° and each neighbour is 180° − 80° = 100°.
- 3.The eight angles use the four measures 60°, 120°, 80° and 100°. Eight angles means eight regions, not necessarily eight different numbers.
For a transversal across parallel lines, corresponding angles repeat between crossings, so there are at most two distinct measures. If the transversal is perpendicular to those lines, all eight angles are 90° and there is only one distinct measure. The maximum of four refers to two arbitrary crossed lines.
Corresponding Means the Same Position
Imagine sliding a small tracing of one crossing to the other while keeping its orientation unchanged. The angle above the first line and to the right of the transversal corresponds to the angle above the second line and to the right of the transversal. The word names a pair of positions, even before we know whether the measures are equal.
A pair of angles at different crossings that occupy the same relative position with respect to the transversal and the two crossed lines. Equal corresponding angles are guaranteed when the two crossed lines are parallel.
| Upper angle | Matching lower angle | Relative position in our diagram |
|---|---|---|
| 1 | 5 | Above crossed line, left of transversal |
| 2 | 6 | Above crossed line, right of transversal |
| 3 | 7 | Below crossed line, right of transversal |
| 4 | 8 | Below crossed line, left of transversal |
Two angles do not become corresponding simply because their measures happen to match. First locate the positions, then apply a rule. Vertically opposite angles share one vertex; corresponding angles have different vertices. This distinction helps avoid moving a known angle to the wrong place in a diagram.
Equal Corresponding Angles and Parallel Lines
The corresponding-angle rule works in two directions. If the two crossed lines are already known to be parallel, we can calculate equal corresponding angles. If we instead know that a pair of corresponding angles is equal, we can conclude that the two lines are parallel.
Draw line l and transversal t crossing at X. Make the upper-right angle 60° with a protractor. Its neighbouring angles are 120°. Mark a different point Y on t. Trace the 60° angle and slide the tracing to Y without rotating, or use a protractor at Y to copy 60° in the corresponding position. Draw line m through Y along the copied side. The new line has the same direction as l, so the two lines are parallel; the eight angles now use only 60° and 120°. Copying a 60° angle on an arbitrary side of t would not necessarily make the intended corresponding pair.
The same relationship holds for any one correctly identified corresponding pair, such as 1/5, 3/7 or 4/8. All the lines are on the same plane. These two directions serve different purposes: one finds angle measures from parallelism; the other establishes parallelism from angle information.
Draw or construct parallel lines l and m, mark them with matching arrows, and add a transversal. Trace one angle, then slide the tracing to the corresponding position at the other crossing without turning it. The sides align. Check the other corresponding pairs with your tracing and a protractor. The measurements illustrate the exact geometric rule; small drawing errors may make them only approximately equal.
Problem
For the numbered diagram, consider two cases: l ∥ m and ∠2 = 67°; or parallelism is unknown but ∠2 = ∠6 = 67°. What follows?
- 1.In the first case, parallelism is given. Since 2 and 6 correspond, ∠6 = 67°.
- 2.In the second case, equality of a corresponding pair is given. Apply the reverse direction to conclude l ∥ m.
- 3.The numerical equality is the same, but the role of the information differs. Always state whether parallelism is given or being established.
Draw two lines l and m that are not parallel. Try different transversals crossing both at distinct points, and measure a correctly identified corresponding pair. You will not obtain exact equality. If you did obtain it, the corresponding-angle test would force l and m to be parallel, contradicting the starting condition. An approximate equality caused by imprecise measurement is not an exact geometric construction.
With nonparallel lines, corresponding pairs still exist, but their measures are not equal. Also, equal angles in unrelated positions alone do not apply this test. Write down the pair and why it is corresponding before concluding parallelism.
A Ruler and Set Square Preserve Direction
A set square is a triangular drawing tool with a right-angle corner. A ruler can act as a fixed guide so that the set square slides without turning. If its orientation stays unchanged, a chosen drawing edge makes the same angle with the guide at each position.
Draw a baseline l. Place one right-angle side of the set square along l and draw along its other right-angle side. Slide the tool along l without rotating and draw again along that same edge. Both new lines are perpendicular to l. With l acting as their transversal, the corresponding angles are both 90°, so the new lines are parallel.
Try the long sloping side of the set square too. Hold a ruler against one side as a guide and slide the set square along it without rotation. Lines drawn along the same sloping edge at different positions make equal corresponding angles with the ruler line, even though the common angle is not 90°. They are therefore parallel. A careless twist changes the angle and defeats the method.
Problem
A point A lies off line l. Describe how to draw a parallel through A using a ruler and set square.
- 1.Place a chosen edge of the set square along l. Put the ruler firmly against another edge of the set square so it becomes a fixed sliding guide.
- 2.Hold the ruler still. Slide the set square along it until the original drawing edge passes through A. Do not rotate the set square.
- 3.Draw m through A along that edge. The edge has kept its direction and its angle with the fixed guide, so corresponding angles are equal and m ∥ l.
A Parallel Through A by Paper Folding
Paper folding provides a second way to preserve a right angle. If we make a crease perpendicular to a given crease, then make another perpendicular to the new crease, we return to the original direction. This connects folding to the same corresponding-angle test used with tools.
Start with a crease l and mark A off it. Fold the paper so that l lies exactly on itself and the new fold passes through A; this crease t is perpendicular to l. One practical way is to align the two portions of l while adjusting the fold to pass through A. Unfold. Now fold so that t lies on itself with the new fold again through A, producing m perpendicular to t. Unfold and inspect all three creases. Since l and m both make 90° with t, the corresponding angles are equal and m ∥ l.
Problem
On one sheet, p ⊥ t and q ⊥ t at different points. Prove p ∥ q.
- 1.Use t as the transversal crossing p and q.
- 2.The corresponding angles at the two crossings are right angles, so they are equal: 90° = 90°.
- 3.The converse corresponding-angle rule gives p ∥ q. The same-plane and distinct-point conditions are part of this reasoning.
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
In the numbered diagram, which angle corresponds to ∠4?
For two arbitrary lines crossed by a transversal, the eight angles have at most how many distinct measures?
A corresponding pair measures 72° and 72°. What can we conclude?
Parallel lines are crossed by a transversal. If ∠1 = 109°, what is ∠5?
Why must a set square slide without rotation when drawing parallels?
Both l and m are perpendicular to t on one plane at different points. What follows?
Both 4 and 8 are below the crossed line and to the left of the transversal.
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
- Name all four corresponding pairs and four vertically opposite pairs in the numbered diagram.
- At one crossing an angle is 35°; at another it is 70°. The crossed lines are not assumed parallel. List the distinct measures.
- In the numbered diagram, ∠2 = 64° and ∠6 = 66°. Can l and m be parallel in the ideal figure?
- Construct parallel lines by copying a 60° corresponding angle at a second point of a transversal. Explain each step.
- Draw a parallel through A with a ruler and set square, then describe why the construction works.
- Draw two lines with the long side of a sliding set square. Must the angle with the ruler be 90° for them to be parallel?
- Create a parallel through A using two perpendicular folds. State the two perpendicular relationships and the final conclusion.
- Can nonparallel lines have a transversal with an exactly equal corresponding pair? Explain without relying on repeated drawings.
Corresponding: 1/5, 2/6, 3/7, 4/8. Vertically opposite: 1/3, 2/4, 5/7, 6/8. Corresponding pairs use different vertices; vertically opposite pairs use one vertex.
Key Takeaways
• A transversal crosses two lines at distinct points and forms two sets of four angles. • At most four distinct measures occur for two arbitrary crossed lines. • Corresponding angles occupy matching positions at different crossings. • Parallel lines give equal corresponding angles; equality of a corresponding pair also establishes parallelism. • Controlled sliding preserves angle and therefore direction. • Two distinct coplanar lines perpendicular to the same line are parallel. • Two perpendicular folds construct a parallel through a chosen point.