🛤️Parallel Lines & Transversals

Two lines that never meet are parallel. A third line that cuts across both of them is a transversal. That single picture — two parallel lines and one line slicing through them — produces eight angles, and almost the whole of this topic is knowing which of those eight are equal and which of them add to 180°.

Stonewright Vela, who cuts stone at Angle Canyon, will not let an apprentice guess. Before any measuring happens, the angles get numbered, and they are always numbered the same way:

Transversal t crosses line p (above) and line q (below). At each crossing the four angles are numbered clockwise, starting from the upper-left.

at line p: at line q: 1 | 2 5 | 6 ----+---- p ----+---- q 4 | 3 8 | 7

So at the top crossing, angle 1 is upper-left, angle 2 is upper-right, angle 3 is lower-right and angle 4 is lower-left. At the bottom crossing, angle 5 is upper-left, angle 6 is upper-right, angle 7 is lower-right and angle 8 is lower-left. Every question in this skill states that convention in its first sentence, so you never have to remember it — but you do have to use it. Point at the picture above with a finger while you read a question, and half the difficulty disappears.

Two words come up constantly. The angles between p and q are the *interior* ones: 3, 4, 5 and 6. The angles outside the two lines are the *exterior* ones: 1, 2, 7 and 8.

Every named pair takes one angle from the top crossing and one from the bottom. Read the positions, and the name follows:

• ◆Corresponding angles sit in *matching* positions at the two crossings — upper-left with upper-left, lower-right with lower-right. So: 1 and 5, 2 and 6, 3 and 7, 4 and 8.
• ◆Alternate interior angles are both between the lines, on *opposite* sides of t: 3 and 5, 4 and 6.
• ◆Alternate exterior angles are both outside the lines, on *opposite* sides of t: 1 and 7, 2 and 8.
• ◆Co-interior angles (also called same-side interior angles) are both between the lines, on the *same* side of t: 3 and 6, 4 and 5.

Two more pairs live at a single crossing and need no parallel lines at all. Vertical angles are opposite each other across a corner and are always equal: 1 and 3, 2 and 4, 5 and 7, 6 and 8. A linear pair sits side by side on a straight line and always adds to 180°: 1 and 2, 2 and 3, 3 and 4, 4 and 1, and the same four at the other crossing.

The memory aid is the shape the pair draws. Corresponding angles make an F (rotate it any way you like). Alternate interior angles make a Z. Co-interior angles make a C or a U. F and Z angles are equal; C angles are supplementary, meaning they add to 180°.

Here is the shortcut that makes the whole picture collapse to one number. When p is parallel to q, only two sizes exist among the eight angles: some angle and its supplement. In this numbering, all the odd-numbered angles are equal to each other, and all the even-numbered ones are equal to each other, and an odd one plus an even one is 180°. If angle 1 is 70°, then angles 3, 5 and 7 are 70°, and angles 2, 4, 6 and 8 are 110°. Use the names to explain your answer; use that shortcut to check it.

The theorems run in two directions, and telling them apart is the real Geometry skill here.

Forwards you are *told* the lines are parallel and you deduce an angle. "Line p is parallel to line q, and angle 2 is 118°. How many degrees is angle 6?" Angle 2 and angle 6 are corresponding, corresponding angles between parallel lines are equal, so angle 6 is 118°. Give angles as a whole number of degrees.

Sometimes the two angles have no single name — angle 2 and angle 5, for instance, are neither corresponding, nor alternate, nor co-interior. Then travel through a third angle in two steps: angle 2 and angle 4 are vertical, so angle 4 matches angle 2; angle 4 and angle 5 are co-interior, so angle 5 is 180° minus that. Naming each step keeps you honest.

Backwards nobody has told you the lines are parallel; you are handed two measures and asked whether they *force* it. Each theorem has a converse: if corresponding angles are equal, then the lines are parallel; if alternate interior angles are equal, then the lines are parallel; if alternate exterior angles are equal, then the lines are parallel; if co-interior angles add to 180°, then the lines are parallel. The reason you give must name the pair you were actually shown — "converse of the alternate interior angles theorem" is the wrong reason for a corresponding pair, even when the lines really are parallel.

Two traps live on this side. First, a fact about two angles at the *same* crossing proves nothing: "angle 1 is congruent to angle 3" is true whatever the lines do, because vertical angles are always equal, and "angle 1 and angle 2 are supplementary" is always true too. A test has to link one angle at p with one angle at q. Second, the test has to match the pair: angles that parallel lines make equal are not the ones that have to add to 180°.

The same two directions turn up in algebra. If angle 2 is (3x + 10)° and angle 6 is (5x - 30)°, those are corresponding angles, so they are equal: 3x + 10 = 5x - 30, giving 2x = 40 and x = 20. If instead the pair were co-interior, you would add the two expressions and set the total to 180. Read the last line of the question before you write anything down: some ask for x, and some ask for the angle, which means putting x back into its expression.

Perpendicular means meeting at exactly 90°. Slot that into the picture and two useful facts fall out.

If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other as well. All four angles at the first crossing are right angles, so by corresponding angles all four at the second crossing are right angles too — every one of the eight angles is 90°.

Run it backwards and you get the fact builders actually use: two lines perpendicular to the same line are parallel to each other. Treat the shared line as a transversal; it makes a 90° angle with each of the other two, those two right angles are corresponding, they are equal, and the converse of the corresponding angles postulate does the rest. It is why Vela can square two walls off the same straight edge and know they will never converge, without ever measuring the gap between them.

A perpendicular line is also a handy measuring stick. If line s meets line q at 22° and line t is perpendicular to q, then the acute angle between s and t is 90° - 22° = 68°. If the angle you are handed is obtuse, take its supplement first: a line meeting q at 124° meets it at 180° - 124° = 56° on the other side, so its acute angle with t is 90° - 56° = 34°.

A steady routine beats a good memory:

1. Say where each angle sits, out loud or on paper: "angle 4 is below line p, on the left of t". 2. Same crossing, or different? Same crossing gives a vertical pair or a linear pair. 3. Different crossings: are both angles between the lines, both outside, or one of each? Same side of t, or opposite sides? 4. Name the pair. Then, and only then, decide whether it is equal or adds to 180°. 5. Check the last line of the question. An angle, or x? A reason, or a number?

Answer formats in this skill. Angles are typed as a whole number of degrees — just the number, so "118", not "118°" and not "angle 6 = 118". A value of x is a whole number too. Anything that asks for the name of a pair, the reason a pair of lines is parallel, the relationship between two lines, or which angle partners another is multiple choice: pick the option, and pick the one whose *reason names the pair you were given*.

One last habit worth building. Before you submit an angle, ask whether it should be equal to the one you were given or supplementary to it, and check that your number looks that way. Half of all lost marks in this topic are a perfectly good calculation of 180° minus the answer.