Before you begin
This part of geometry has almost no arithmetic in it. Nearly every angle question on the GED is answered by subtracting from 180, or from 90, or from 360, and the hard part is never the subtraction. The hard part is knowing which of those three numbers you are subtracting from, and that comes from reading the picture.
So this quiz is mostly pictures. There are only about six facts in the whole family, and each one is a picture you can redraw on scratch paper in four seconds. Once you can draw them, the questions answer themselves.
What an angle measures
An angle measures turn. It does not measure length, and this is the first thing to get straight, because the drawings fight you on it.
Stand facing a wall and turn all the way round until you face it again: that is 360 degrees, a full turn. Turn halfway, so you face the opposite wall: 180 degrees. Turn a quarter, so you face the side wall: 90 degrees.
Now the part that misleads people. If you draw an angle with very long sides and the same angle again with very short sides, they are the same angle. Nothing about how far the lines run changes how much turn there is between them. A test question will sometimes draw one angle with long arms and another with stubby ones, and the eye insists the long one is bigger. It is not. Only the opening counts.
Those four names are worth knowing because questions use them as clues. If a problem says an angle is obtuse and you have worked out 40 degrees, you have made a mistake somewhere and the word just told you so.
Two angles on a line add to 180
Draw a straight line. Put a point on it and draw a ray up from that point. You have cut the straight angle into two, and the two must still add to the whole: 180.
Two angles that add to 180 are called supplementary. Two that add to 90 are called complementary. The GED does use both words, and they are easy to mix up. One way to keep them apart: c comes before s in the alphabet, and 90 comes before 180. Complementary is the smaller pair.
Either way, the work is the same. Given one, subtract: 180 − 118 = 62, or 90 − 37 = 53.
Where two lines cross
Two straight lines crossing make four angles. It looks like four things to work out. It is one.
The two facts are worth saying separately, because they are the two you use.
Angles opposite each other are equal. They are called vertical angles, which is an unhelpful name — it has nothing to do with up and down. It means the pair that touch only at the point, across the crossing from each other.
Angles next to each other add to 180, because each such pair sits along one of the straight lines.
Put the two together and the whole picture holds only two different numbers, and those two add to 180. So one angle is enough. Told that one of the four is 74, you have them all at once: 74, 106, 74, 106.
Two parallel lines, crossed by a third
Now the picture the test likes most. Two parallel lines — two lines running in the same direction, never meeting — are cut by a third line running across them. That makes eight angles.
Eight angles, and the same thing is true as before, only more so: there are still only two different numbers in the picture. Four of the angles are one size, four are the other, and the two sizes add to 180.
You can see which is which without any vocabulary at all. Every angle that looks sharp is one number. Every angle that looks wide is the other. Since the two lines are parallel, the lower crossing is an exact copy of the upper one, slid down the page.
So if angle 1 is 65 degrees, then angles 4, 5 and 8 are also 65, and angles 2, 3, 6 and 7 are all 115.
The three angles of a triangle add to 180
This is the most useful fact in the family and the one the test leans on hardest.
Here is why it is true, and it is worth doing once with real paper. Cut a triangle out — any triangle, drawn carelessly. Tear off its three corners. Now lay the three torn corners side by side with their points touching. They make a straight line.
So two angles always give you the third. A triangle with angles of 43 and 68: add them, 111, and take that from 180. The third is 69.
Three cases follow from this and each one saves you a step.
A right triangle. One angle is already 90, so the other two must share the remaining 90 between them. Given one of them as 34, the last is 56, and you never touch 180.
An isosceles triangle has two equal sides, and the two angles opposite those sides are equal as well. If the odd angle at the top is 40, the two at the bottom share 140 and get 70 each.
An equilateral triangle has three equal sides, so all three angles are equal, so each is 60. Always 60, in every equilateral triangle that has ever been drawn.
The angle on the outside
Extend one side of a triangle past a corner and you make a new angle outside it, called an exterior angle.
There are two ways to find it, and they always agree. The exterior angle and the interior angle beside it sit on a straight line, so 180 − 70 = 110. Or use the rule: an exterior angle equals the two interior angles it does not touch, added. Here 50 + 60 = 110.
The rule is not a separate fact. The three interior angles add to 180, and the interior angle at that corner plus the exterior angle also add to 180, so the exterior angle has to equal what is left — the other two. It is the same 180 counted twice. Knowing both routes is useful because a question sometimes hands you the two far angles and not the near one.
Four-sided shapes add to 360
The four angles of any four-sided figure add to 360. Not only squares and rectangles: any four-sided shape, however lopsided.
The reason is the triangle fact again. Draw a line from one corner of a four-sided shape to the corner across from it and you have cut it into two triangles. Two triangles, 180 each, 360 altogether.
That also tells you how to handle a five- or six-sided shape if one ever appears: cut it into triangles from one corner and count them. Five sides make three triangles, so 540.
The guard
Three things to hold onto.
Do not measure with your eye. Test figures carry the words not drawn to scale more often than not, and even when they do not, they are not reliable. An angle drawn looking like 90 may be 80. Work from the facts and the numbers given, never from how the picture looks. The one thing the picture is always honest about is which angles are which — what touches what, and what sits opposite what.
Write on the figure. Every time you work out an angle, write it into the drawing. Angle problems come apart one number at a time, and each number you fill in makes the next one visible. Trying to hold four angles in your head is what makes these questions feel hard; they are not hard on paper.
Ask which total you are in. Before subtracting, name the number you are subtracting from. On a straight line, 180. In a right angle, 90. In a triangle, 180. Around a four-sided shape, 360. All the way around a point, 360. Almost every wrong answer in this family is the right subtraction from the wrong total.
Three to study before you start
Two straight lines cross. One of the four angles measures 118°. Find the other three.
The angle opposite the 118 is equal to it, so that one is 118°. Each of the other two sits on a straight line with the 118, so each is 180 − 118 = 62°.
Reading round the crossing: 118, 62, 118, 62. Two numbers, alternating, adding to 180 in every neighboring pair. If you ever get four different numbers at a crossing, something has gone wrong.
A triangle has angles of 43° and 68°. What is the third?
Add the two you have: 43 + 68 = 111. Take that from 180: 180 − 111 = 69°.
Check it by adding all three: 43 + 68 + 69 = 180. That check takes three seconds and catches the most common slip in this family, which is subtracting only one of the two given angles and answering 137 or 112.
Two parallel lines are crossed by a third. One of the eight angles is 72°. What are the rest?
Four of the eight are 72° and the other four are 180 − 72 = 108°. Which four are which you can read straight off the drawing: every sharp-looking angle is 72, every wide one is 108.
You do not need to decide whether a particular pair is corresponding or alternate interior. Those names describe why two angles are equal; they do not change which ones are. Sharp with sharp, wide with wide, and the two add to 180.
Now you
Work on paper. No calculator is needed anywhere here. Draw the picture for any question that does not come with one, and write each angle into your drawing as you find it. Questions 6 and 10 have two parts.
- Two angles sit together on a straight line. One measures 118°. What does the other measure?
- Two angles together make a right angle. One measures 37°. What does the other measure?
- Two straight lines cross. One of the four angles measures 74°. What are the other three?
- A triangle has angles of 52° and 61°. What is the third angle?
- A) 67°
- B) 113°
- C) 119°
- D) 128°
- A right triangle has one angle of 34°, not counting the right angle. What is its third angle?
- In the figure below, two parallel lines are crossed by a third. Angle 1 measures 65°.
(a) Which other angles measure 65°?
(b) What do the remaining four angles measure?
Reading A reading rest stop, for the stubborn ones.
Ten minutes with the last section of this page — The Company, below the answer key — before you finish the quiz. It is about a librarian who measured the whole Earth using a stick, a shadow, and exactly the fact about parallel lines that you have just learned. It will not help you with question 7. It may be the best argument anyone has made for why this picture is worth knowing.
No photograph at this rest stop. There is a drawing down there instead, and it is the same figure as the one at the start of the Guide, wrapped around a planet.
- An isosceles triangle has a top angle of 40°. What are its two base angles?
- Tonia says a triangle can have two right angles. Is she right? Say why or why not.
- A four-sided figure has angles of 100°, 85° and 95°. What is the fourth angle?
- A) 80°
- B) 90°
- C) 100°
- D) 280°
- A triangle has two angles measuring 50° and 60°. (a) What is the third angle? (b) One side is extended past that third corner. What is the exterior angle made there?
Check your work
The man who measured the Earth with a stick
Around 240 BC, a librarian in Alexandria worked out how big the world is. He was not far wrong, and the whole calculation rests on the fact you have just spent this quiz learning.
His name was Eratosthenes. He ran the Library at Alexandria, which was the largest collection of writing in the ancient Mediterranean, and he had a reputation among his contemporaries for being second-best at everything — a decent poet, a decent mathematician, a decent geographer, first-rate at none of them. They nicknamed him Beta, the second letter. He is the one we still read.
What he knew was this. Far to the south, at a town called Syene — Aswan today — there was a well, and at noon on the longest day of the year the sun shone straight down into it and lit the water at the bottom. A pole standing upright at Syene at that moment cast no shadow at all. The sun was directly overhead.
At Alexandria on the same day at the same hour, an upright pole did cast a shadow. Eratosthenes measured the angle between the pole and the sunlight, and found it was about a fiftieth of a full circle — 7.2 degrees.
Here is the step that makes it work. The sun is so far away that its rays arriving at the two towns are, for practical purposes, parallel. An upright pole points straight down at the center of the Earth, so the two poles are two radii of the same circle, extended. That is exactly the figure at the start of the Guide: two lines crossed by a third, with the sunlight as the third line.
And so the angle Eratosthenes measured at Alexandria — between the pole and the sunlight — is equal to the angle at the center of the Earth between Alexandria and Syene. The same alternate angles, on the scale of a planet. He could not go to the center of the Earth and measure it. He did not need to. He measured a shadow instead.
If the angle between the two towns is a fiftieth of the way around the Earth, then the distance between the two towns is a fiftieth of the way around the Earth. The distance was known: about 5,000 stadia. Multiply by 50 and you get 250,000 stadia for the whole circumference. He later gave the figure as 252,000, which is neater to divide.
How close was he? Honestly, nobody can say exactly, and it is worth being straight about why. We do not know which stadion he was using, and different ones were in use. On the shorter of the plausible values his answer is within a few percent of the real figure, which is 40,008 kilometers around the poles. On the longer one he is out by more than a tenth. The method is not in doubt; the unit is. What is certain is that in 240 BC a man with a pole and some arithmetic got an answer of the right size, when the alternative was having no idea at all.
There is one more part of this that usually gets left out, and it is the part that belongs here.
Eratosthenes did not measure the distance from Alexandria to Syene. He could not have; it is several hundred miles up the Nile. That number came from other people's labor. The Ptolemaic state employed bematists — professional pacers, men trained to walk long distances at a controlled stride and count their steps — to survey the roads of the kingdom. Their counts, and the travel times of the camel caravans, are where the 5,000 stadia came from.
So the measurement of the Earth is two things joined. A clever idea about angles, from a librarian who never left the city. And a very large number of steps, counted by men whose names are not recorded, walking south along a river in the heat. Neither half works without the other. The geometry on this page is the free part, available to anyone with a stick; the rest of it was somebody's day.