Nine names, three facts

This topic hands you a long list of names: complementary, supplementary, adjacent, vertically opposite, corresponding, alternate, co-interior. It looks like seven separate things to learn. It is not. Every single one of them is one of three simple facts wearing a different label, and once you see which three, the list stops being a list.

The three facts

A straight line is 180°. Anything sitting along a straight line has to add up to 180.
A full turn is 360°. Anything going right round a point has to add up to 360.
Parallel lines slide an angle along unchanged. Cross two parallel lines with one line and the same angle reappears at the second crossing.

The names that come from a straight line

Name	What it means	Which fact
Complementary	Two angles that add up to 90°	Half of a straight line
Supplementary	Two angles that add up to 180°	A straight line
Adjacent	Two angles side by side sharing an arm and a corner	Only a description, not a sum
Linear pair	Adjacent angles whose outer arms make a straight line, so they add to 180°	A straight line
Vertically opposite	The two angles facing each other where two lines cross. They are equal	A straight line, used twice

Vertically opposite angles, proved in one line

Two lines cross and make four angles. Call them a, b, c and d going round. Now a and b sit on a straight line, so a + b = 180. And b and c also sit on a straight line, so b + c = 180. Both equal 180, so a + b = b + c, and taking b off both sides leaves a = c. That is the proof. It uses the straight line fact twice and nothing else, and it explains why nobody has to measure vertically opposite angles to know they are equal.

The names that come from parallel lines

Draw two parallel lines and cut across both with a third line. That third line is called a transversal. Eight angles appear, four at each crossing. Here is the thing worth understanding: because the two lines are parallel, the second crossing is an exact copy of the first, simply slid along. So there are really only two different angle sizes in the whole picture, and every one of the eight angles is one or the other.

Name	Where they sit	Relationship
Corresponding	Same position at each crossing, like the top left of both	Equal
Alternate	Opposite sides of the transversal, between the two parallel lines	Equal
Co-interior	Same side of the transversal, between the two parallel lines	Add up to 180°

How to get all three without memorising them

Pick any angle in the picture. Slide it along to the other crossing, keeping it in exactly the same position. That is the corresponding angle, and sliding does not change size, so they are equal. Now from that one, use vertically opposite to jump across the crossing, which also does not change size. That gives you alternate angles, equal again. Finally, take an alternate angle and swap it for its neighbour on the straight line, which turns equal into adds to 180. That gives you co-interior. Three names, one slide and two moves you already know.

The condition everyone forgets

Corresponding, alternate and co-interior only work if the two lines really are parallel. If they are not, none of the three relationships holds and the eight angles can be almost anything. Exam questions exploit this constantly: they draw two lines that look parallel, do not mark them as parallel, and wait to see who assumes. Look for the arrow marks on the lines. No arrows, no assumption. The straight line and full turn facts, by contrast, are always true and never need any condition.

Complementary and supplementary are easy to mix up

Both words start similarly and mean different numbers. C comes before S in the alphabet, and 90 comes before 180. Complementary is the smaller one at 90, supplementary is the larger one at 180. Use that once and you will not lose a mark on it again. A second way to remember: a Corner is 90 degrees and a Straight line is 180.

Worked Example 1

Two lines cross. One angle is 68°. Find the other three.

Angle	Reason	Value
a	Given	68°
b	a and b make a straight line, so b = 180 - 68	112°
c	Vertically opposite to a	68°
d	Vertically opposite to b	112°

CheckAll four should add to 360, because they go right round a point. 68 + 112 + 68 + 112 = 360. Correct. Doing this check costs five seconds and catches most arithmetic slips.

Worked Example 2

Two parallel lines are cut by a transversal. One angle is 115°. Find all eight.

Think firstThere are only two different sizes in the whole diagram, 115 and its partner on the straight line, which is 180 - 115 = 65. So every one of the eight angles is either 115 or 65. The only work left is deciding which is which.

The quick ruleAngles in the same position at both crossings are equal. Angles next to each other at the same crossing add to 180. Working round the first crossing: 115, 65, 115, 65. Then copy that exact pattern onto the second crossing, because parallel lines slide it along unchanged.

The sentence that earns the marksWrite the reason next to every value, not just the value. For example: co-interior angles are supplementary, so the angle is 180 - 115 = 65. A page of correct numbers with no reasons scores far less than a page with reasons, because the question is testing whether you know why.

Activity 1.3

Verify the facts with a protractor

MethodDraw two lines crossing at any angle you like. Measure all four angles with a protractor and write them down. Add the four together. Then find the two pairs of vertically opposite angles and compare them. Repeat with a completely different crossing angle.

What you should findThe four always add to 360, and the two opposite pairs are always equal, no matter what crossing angle you chose. Measuring does not prove this, though. It only makes it believable. The proof is the two line argument in the Note tab, and that is what an exam wants.

Group work

Find the parallel lines in this room

MethodIn groups, list ten places in the classroom or outside where two parallel lines are cut by a third line: the lines on a ruled page crossed by a margin, the rails of a ladder crossed by its rungs, floor tiles, a window grill, the strings of a charpai. For each one, point out a pair of corresponding angles and a pair of alternate angles.

The harder halfNow find something that looks parallel but is not, and check it. Railway lines drawn in a picture appear to meet in the distance. The sides of a road do the same. Ask what the arrow marks would look like on those, and why a drawing can lie about parallelism when the real thing does not.

Worked Example

One group did the second half properly and came back with something their teacher had not expected. Here is their table.

Reflection

The bench row is the one that taught us something. It looked parallel and we would have marked it parallel if we had not measured. Two centimetres over a metre is invisible to the eye but it is not parallel, and if we had used corresponding angles on it our answers would have been slightly wrong with no way of knowing. That is exactly why a diagram has to be marked and not just drawn.

Your TurnFind four things and check them properly, by measuring the gap at both ends rather than by looking.

What I looked at	Gap at one end	Gap at the other	Parallel?
-	-	-	-
-	-	-	-
-	-	-	-
-	-	-	-

Every angle-pair name in this topic comes from one of three facts: a straight line is 180, a full turn is 360, parallel lines slide an angle along unchanged.

Complementary angles add to 90°. Supplementary angles add to 180°. C before S, 90 before 180.

Adjacent angles share a corner and an arm. That is a description, not a sum.

A linear pair is adjacent angles whose outer arms make a straight line, so they add to 180°.

Vertically opposite angles are equal. Proof: a + b = 180 and b + c = 180, so a = c.

The four angles where two lines cross always add to 360°, because they go right round a point.

A transversal is the line that cuts across two other lines.

Corresponding angles sit in the same position at each crossing and are equal.

Alternate angles sit on opposite sides of the transversal, between the lines, and are equal.

Co-interior angles sit on the same side of the transversal, between the lines, and add to 180°.

With two parallel lines and a transversal there are only two different angle sizes in the whole diagram.

Corresponding, alternate and co-interior only hold if the lines are truly parallel. Look for the arrow marks.

In an exam, write the reason beside every value. Numbers without reasons score very little.

1Explain why vertically opposite angles are equal, without measuring anything.

Let two lines cross and label the four angles a, b, c and d in order round the crossing point. The angles a and b lie together along one straight line, so a + b = 180. The angles b and c lie together along the other straight line, so b + c = 180. Since both expressions equal 180, they equal each other: a + b = b + c. Subtracting b from both sides gives a = c. The same argument applied to the other pair gives b = d. Notice that this uses only the fact that a straight line is 180 degrees, applied twice, and it needs no parallel lines and no measurement at all.

2What is the difference between complementary and supplementary angles?

Complementary angles are two angles that add up to 90 degrees, which is a right angle or the corner of a square. Supplementary angles are two angles that add up to 180 degrees, which is a straight line. The two words are easy to confuse because they sound similar, so it helps that C comes before S in the alphabet just as 90 comes before 180. Another way to hold on to it is that a Corner is 90 and a Straight line is 180. Neither pair has to be next to each other in the diagram: two angles in completely different places can still be complementary if their sizes add to 90.

3Two parallel lines are cut by a transversal and one angle is 72°. Find all eight angles and give a reason for each kind.

There are only two sizes in the whole diagram. The given angle is 72, and its neighbour on the straight line is 180 minus 72, which is 108. So every one of the eight angles is either 72 or 108. Going round the first crossing the sizes alternate 72, 108, 72, 108, because each neighbouring pair forms a linear pair adding to 180. The second crossing carries exactly the same pattern in the same positions, because corresponding angles are equal when the lines are parallel. Reasons to write down: neighbours at a crossing are supplementary; angles facing each other are vertically opposite and equal; angles in matching positions at the two crossings are corresponding and equal; angles on opposite sides of the transversal between the lines are alternate and equal; angles on the same side between the lines are co-interior and add to 180.

4A diagram shows two lines cut by a transversal but no arrow marks. What can and cannot be concluded?

You can still use everything that comes from a straight line and a full turn, because those facts are always true. So at each crossing the neighbouring angles are supplementary, the opposite angles are equal, and all four add to 360. What you cannot do is relate one crossing to the other. Corresponding, alternate and co-interior all depend on the two lines being genuinely parallel, and without arrow marks you have not been told that they are. Two lines can look parallel to the eye and be a degree or two out, which is invisible in a drawing but enough to make every one of those three relationships false. Examiners use this on purpose to separate students who check from students who assume.

5Why does the whole eight angle picture only contain two different sizes?

Because the second crossing is a copy of the first. When a line cuts two parallel lines, it meets them at exactly the same angle both times, since the lines never converge or diverge. So the pattern of four angles at the first crossing reappears unchanged at the second. And at any single crossing there are only two sizes anyway, because neighbouring angles are supplementary and opposite angles are equal: the pattern round one crossing is x, 180 minus x, x, 180 minus x. Put those two observations together and the whole eight angle diagram contains just x and 180 minus x. This is why these questions are far easier than they look: find one angle and you have found all eight.

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The video for this topic is being prepared. Check back soon!

Unit 1 · Lines and Angles

Pairs of Angles

Mathematics · Grade 7

Three Facts, Nothing Else

📏

A straight line is 180°. Always true, no conditions

🔄

A full turn is 360°. Always true, no conditions

➡️

Parallel lines slide an angle along unchanged. Only if truly parallel

🏷️

Every name in this topic is one of these three with a label on it

Vertically Opposite, Proved in Four Lines

1Two lines cross. Label the four angles a, b, c, d going round

2a and b lie on one straight line, so a + b = 180

3b and c lie on the other straight line, so b + c = 180

4Both equal 180, so a + b = b + c. Take b off both sides

5a = c. Nothing was measured and no lines had to be parallel

“

Because the two lines are parallel, the second crossing is an exact copy of the first, simply slid along. So there are really only two different angle sizes in the whole eight angle picture.

Three Names From One Slide

1Pick any angle in the picture and call it x

2Slide it to the other crossing, same position. CORRESPONDING, equal

3Now jump across that crossing. Vertically opposite, so still equal. ALTERNATE

4Swap one for its neighbour on the straight line. CO-INTERIOR, adds to 180

Always True, or Only If Parallel

✅ Always true

Angles on a straight line add to 180

Angles round a point add to 360

Vertically opposite angles are equal

⚠️ Only if the lines are parallel

Corresponding angles are equal

Alternate angles are equal

Co-interior angles add to 180

No arrow marks on the lines? Then only the green column is available

GROUP WORK · 25 MIN

Find the Parallel Lines, Then Find the Liars

HUNT · 10 MIN

List ten places where two parallel lines are cut by a third: ruled paper, a ladder, a window grill, floor tiles

NAME · 8 MIN

On each one, point out a pair of corresponding angles and a pair of alternate angles

CHECK · 7 MIN

Now measure the gap at BOTH ends of each pair. Find at least one thing that looked parallel and is not

How to Answer These Questions

1️⃣

Find any one angle. In a parallel line picture that gives you all eight

✍️

Write the reason beside every value, not just the number

🔍

Check for arrow marks before using anything from the parallel family

➕

Check at the end: the four angles at a crossing must add to 360

“

Two lines can look parallel to the eye and be a degree out. That is invisible in a drawing and it is enough to make corresponding, alternate and co-interior all false. Which is why a diagram has to be marked, not just drawn.

Presenter notes: Open by writing all seven names on the board in a long column: complementary, supplementary, adjacent, linear pair, vertically opposite, corresponding, alternate, co-interior. Let the class look at it and feel how much there is to learn. Then say that by the end of the lesson that column will have collapsed into three lines, and that nobody is going to memorise the list. This framing matters, because a student who believes there are eight separate things to learn will study it as vocabulary, and a student who believes there are three will study it as reasoning.