Chapter 4.4 — Exercise 4.3 — Parallel Lines
Lines and a transversal, lines parallel to the same line. This is Lesson 4 of 5 in Chapter 4: Lines and Angles.
One Line Crossing Two Others
A transversal is a line that crosses two other distinct lines at two distinct points, producing 8 angles total — 4 at each crossing. Exercise 4.3 is entirely about naming the relationships among those 8 angles, and what changes once the two lines being crossed happen to be parallel — the same 8 angles, but with five new numeric guarantees layered on top of them.
Eight Angles, Five Named Pairings
| Pairing | Angle pairs | Position |
|---|---|---|
| Corresponding | (1,5) (2,6) (3,7) (4,8) | Same corner at each crossing |
| Alternate interior | (3,5) (4,6) | Between the lines, opposite sides of l |
| Alternate exterior | (1,7) (2,8) | Outside the lines, opposite sides of l |
| Co-interior | (4,5) (3,6) | Between the lines, same side of l |
| Co-exterior | (1,8) (2,7) | Outside the lines, same side of l |
What Changes Once the Two Lines Are Parallel
Every pairing above is just a position, true regardless of whether m and n are parallel. Once m ∥ n is established, each of these five pairings gains a specific numeric relationship that holds without exception:
- Corresponding angles become equal: ∠1=∠5, ∠2=∠6, ∠3=∠7, ∠4=∠8.
- Alternate interior angles become equal: ∠3=∠5, ∠4=∠6.
- Alternate exterior angles become equal: ∠1=∠7, ∠2=∠8.
- Co-interior angles become supplementary: ∠4+∠5=180°, ∠3+∠6=180°.
- Co-exterior angles become supplementary: ∠1+∠8=180°, ∠2+∠7=180°.
Each of these five also runs in reverse as a test for parallelism itself: if a transversal crosses two lines such that one pair of corresponding angles turns out equal, or one pair of co-interior angles turns out supplementary, that alone is enough to conclude the two lines are parallel — no other information needed. Only one pair from any of the five categories needs to check out for the conclusion to hold; there's no requirement to verify all four corresponding pairs, say, before declaring the lines parallel, since a genuine geometric theorem guarantees the rest follow automatically once any single pair is confirmed. This reverse direction is what makes the properties genuinely useful in constructions, not just in calculations — it's how a carpenter or draftsperson checks that two edges are truly parallel using nothing more than a single angle measurement against a straightedge laid across both.
Chaining Three Parallel Lines
Given AB ∥ CD and CD ∥ EF, the transitive property gives AB ∥ EF directly. If y:z = 3:7 for the angles this creates, with x and y co-interior (x+y=180°) and x and z alternate interior (x=z), substituting gives y+z=180°. Splitting 180° in ratio 3:7 gives y=54° and z=126°, so x=z=126°.
y+z=180°, y:z=3:7 → y=54°, z=126° → x=z=126°Two separate properties had to combine before the ratio could even be applied: recognising AB ∥ EF required the transitive property first, and only after that did the co-interior and alternate-interior relationships become available to use at all. Skipping the transitive step — trying to relate x directly to y or z without first establishing AB ∥ EF — would leave both of those angle properties inapplicable, since they only hold between lines already known to be parallel.
The Auxiliary Line Technique
Several harder problems place a vertex between two parallel lines, touching neither one directly. The fix is to draw a third line through that vertex, parallel to both — splitting one otherwise unreachable unknown angle into two separately measurable pieces. With PQ ∥ ST, ∠PQR=110°, ∠RST=130°: drawing line l through R parallel to both gives a+110°=180° (co-interior with PQ), so a=70°, and c+130°=180° (co-interior with ST), so c=50°. Since a+b+c=180° (angles on a straight line at R), b=180°−70°−50°=60°, giving ∠QRS=60°.
a=70°, c=50°, a+b+c=180° → b=60° = ∠QRSRecognising when the auxiliary-line trick is even needed comes down to one check: does the unknown angle's vertex sit on either of the two parallel lines, or does it sit strictly between them? A vertex on PQ or ST could apply the standard co-interior or alternate-interior properties directly against a transversal. A vertex like R, touching neither line, has no transversal relationship to lean on until a new line is drawn through it — which is precisely what the auxiliary line supplies.
The same auxiliary-line trick handles a general version of this setup: for any point C strictly between two parallel lines m and n, with A on m and B on n, drawing a line through C parallel to both splits ∠ACB into two alternate-interior pairs, giving ∠ACB as the sum of the two angles A and B make with lines m and n respectively — without ever needing to know exactly where C sits between the two lines. The auxiliary line only ever needs to be parallel to the two given lines; its exact position doesn't matter, since it's not a measurement being added to the figure so much as a temporary scaffold for applying alternate-interior or co-interior properties that wouldn't otherwise have anywhere to attach.
Reading Angle Relationships Off a Figure
Many problems just ask which angles are equal to a given one, once two pairs of parallel lines are marked with arrowheads. With a∥b and c∥d — two separate pairs of parallel lines crossing each other, four intersections in total, 16 angles altogether — every angle occupying the same relative position as ∠1 at any of the four crossings is congruent to it: seven others in total (∠3, ∠5, ∠7, ∠9, ∠11, ∠13, ∠15), leaving the remaining eight angles (including ∠2) all congruent to ∠2 instead, since ∠1 and ∠2 form a linear pair and every angle in the figure ends up in one of only these two congruence classes.
Turning a Figure Into an Equation
Numeric problems combine these properties with ordinary algebra. Given 2x+15 and 3x−20 as a pair of corresponding angles under AB ∥ CD, setting the two expressions equal to each other gives 2x+15=3x−20, which rearranges to x=35. Given 4x−23 and 3x as co-interior angles instead, their sum must equal 180° exactly: 4x−23+3x=180, so 7x=203 and x=29. The property used — equal versus supplementary — always depends entirely on which named pairing the two marked angles form, never on the specific numbers involved. Getting that identification right is really the whole exercise in miniature: two angles marked with the exact same style of arrow at the exact same relative position are corresponding and equal; two angles both tucked between the parallel lines but on the same side of the transversal are co-interior and supplementary — the algebra afterward is only ever as reliable as that initial identification.
Into Triangles Next
Every parallel-line property proved here becomes a tool rather than a topic in its own right once Exercise 4.4 proves the triangle angle sum property — the proof itself works by drawing a line through one vertex parallel to the opposite side, then reading off alternate interior angles exactly as practiced here in this exercise's own auxiliary-line problems. The same transversal properties reappear when Class 10's Similar Triangles chapter proves the Basic Proportionality Theorem, and again in Coordinate Geometry when parallel lines are identified by comparing slopes instead of angles — a different-looking test that turns out to be checking exactly the same underlying fact.