Acute, right, obtuse and straight angles
Angles come in every size from a sliver up to a half turn. Which sizes are worth a name, so that you can tell two angles apart without measuring either?
Where you are 8 in Angle types and pairs
Why this exists
Once you can measure an angle you can give it a number. Most of the time you do not need the number. A carpenter checking a corner or a reader glancing at a diagram wants to know something simpler first: is this angle smaller than a square corner, exactly a square corner, or bigger?
Two sizes act as landmarks for that question. One is the square corner, the right angle. The other is the angle made by a straight line, the straight angle, which is two right angles side by side. Naming the landmarks, and the two ranges on either side of the first one, gives four words that sort every angle between two rays. They are the words the rest of geometry is written in: complementary angles are defined by the right angle, supplementary angles by the straight angle, and triangles are sorted by whether they contain a right or an obtuse angle.
The right angle came first, long before degrees. Euclid defines it without any number at all: when one line stands on another and makes the two angles beside it equal, each of those equal angles is right. Degrees came later and put the landmark at 90^\circ.
Definition
An angle is sorted by its measure, in degrees. For an angle \angle A:
- \angle A is acute when 0^\circ < m\angle A < 90^\circ. It is narrower than a square corner.
- \angle A is right when m\angle A = 90^\circ. It is exactly a square corner, and a drawing marks it with a small square at the vertex instead of an arc.
- \angle A is obtuse when 90^\circ < m\angle A < 180^\circ. It is wider than a square corner but its arms do not yet form a line.
- \angle A is straight when m\angle A = 180^\circ. Its two arms point in opposite directions and together make one straight line through the vertex.
Acute means sharp and obtuse means blunt. A right angle is “right” in the old sense of upright: the angle an upright post makes with level ground.
Notice the shape of the definition. Right and straight are single values, so an angle is right only if its measure is exactly 90^\circ. Acute and obtuse are ranges, with the right angle as the fence between them and belonging to neither.
Intuition
Fix one arm and let the other swing open from on top of it, the way a door swings away from its frame.
At first the angle is thin, and it is acute. It stays acute the whole time the moving arm is in the shaded quarter. The instant the arm stands upright over the fixed one, the angle is right. Swing any further and it is obtuse, and it stays obtuse until the moving arm points exactly away from the fixed one. At that moment the two arms line up into a single straight line and the angle is straight.
A sheet of paper carries the landmark for you. Its corner is a right angle, so holding a corner against an angle tells you at once which side of 90^\circ it is on. That is how angles are estimated before they are measured. The length of the arms plays no part: a tiny acute angle and one drawn with arms a meter long are the same kind, because size is the amount of turn, not the length of the arms.
Results
Every angle has exactly one kind. An angle between two rays with measure m, where 0^\circ < m \le 180^\circ, is exactly one of acute, right, obtuse or straight.
The landmarks are fixed sizes. All right angles are congruent, and all straight angles are congruent.
A ray splits a straight angle into a matched pair. If a ray leaves a point on a line and does not lie along the line, it divides the straight angle there into two angles whose measures add to 180^\circ, and either
Why it is true
The first result is a matter of where a number sits. Compare m with 90: it is less, equal or greater, and exactly one of those holds. If it is greater, it is either less than 180 or equal to 180, and again exactly one holds. So the four ranges
cover every measure from just above 0^\circ up to 180^\circ, and no measure is in two of them.
The second result is the definition read the other way. Every right angle measures 90^\circ and every straight angle measures 180^\circ, and angles with equal measures are congruent. Proving it formally is its own short proof.
For the third, let the ray make an angle of x^\circ with one side of the line. Because the ray is off the line, 0 < x < 180. The two angles fill the straight angle together, so by the angle addition postulate the other one is (180 - x)^\circ, which is also between 0 and 180. Now there are three cases.
If x < 90, then 180 - x > 180 - 90 = 90. One angle is acute and the other obtuse.
If x = 90, then 180 - x = 90. Both are right.
If x > 90, then 180 - x < 90. Again one is obtuse and the other acute.
The middle case is exactly Euclid’s definition. A line standing on a line makes two angles that add to 180^\circ, and they are equal precisely when each is 180^\circ \div 2 = 90^\circ. So “the two equal angles beside a line” and “90^\circ” are two descriptions of one angle, which is why the old definition and the modern one agree.
Worked examples
Sorting by measure
Name the kind of each angle: 35^\circ, 90^\circ, 91^\circ, 179.5^\circ, 180^\circ and 89^\circ.
SolutionCompare each measure with the two landmarks, 90^\circ and 180^\circ.
35^\circ is more than 0^\circ and less than 90^\circ, so it is acute.
90^\circ is exactly the right-angle landmark, so it is right.
91^\circ is past 90^\circ but short of 180^\circ, so it is obtuse. Being only one degree past the landmark makes no difference; obtuse is a range.
179.5^\circ is obtuse for the same reason. It is very nearly straight, but a straight angle must be exactly 180^\circ.
180^\circ is straight.
89^\circ is less than 90^\circ, so it is acute, however square it looks on a page.
Splitting a straight angle
In Figure 3, A, O and B lie on one line and m\angle AOX = 38^\circ. Find m\angle XOB and name the kind of both angles.
\overrightarrow{OA} and \overrightarrow{OB} point in opposite directions along one line, so \angle AOB is a straight angle and measures 180^\circ. The ray \overrightarrow{OX} lies inside it, so the two smaller angles add to the whole:
\angle AOX is acute, since 38^\circ < 90^\circ, and \angle XOB is obtuse, since 90^\circ < 142^\circ < 180^\circ. One acute and one obtuse is exactly what the third result promised.
A measure written in terms of x
An angle measures (3x + 15)^\circ. Find the value of x that makes it a right angle and the value that makes it a straight angle. Then find every value of x that makes it acute.
SolutionA right angle measures exactly 90^\circ, so set the expression equal to 90:
A straight angle measures exactly 180^\circ:
An acute angle is a range, so the condition is two inequalities at once: the measure must be more than 0 and less than 90.
Subtract 15 from all three parts, which keeps the order, then divide all three by 3, which keeps it too because 3 is positive:
So the angle is acute when -5 < x < 25. A negative x is allowed; only the measure has to be positive. For example x = -2 gives 3(-2) + 15 = 9, a 9^\circ angle. The same reasoning shows the angle is obtuse when 25 < x < 55.
The hands of a clock
Find the angle between the hands of a clock at five o’clock, and name its kind. Then do the same at three, six and ten o’clock. In each case take the smaller of the two angles the hands make.
The twelve hour marks divide a full turn of 360^\circ into twelve equal gaps, so each gap is
On the hour, the minute hand points at 12 and the hour hand points exactly at the hour. At five o’clock the hour hand is five gaps round from 12, as Figure 4 shows:
which is obtuse. At three o’clock it is 3 \times 30^\circ = 90^\circ, a right angle, which is why the hands at three look like a square corner. At six o’clock it is 6 \times 30^\circ = 180^\circ: the hands point in opposite directions and make a straight angle.
At ten o’clock, counting forward from 12 gives 10 \times 30^\circ = 300^\circ, which is the larger way round. The smaller angle goes the other way, back two gaps:
so at ten o’clock the hands make an acute angle. Reading the time off an analog clock is, underneath, reading an angle.
Halving and doubling
An angle is cut exactly in half by a ray through its vertex. What kind are the two halves if the original angle is obtuse? If it is straight? And what kinds can you get by doubling an acute angle?
SolutionIf the original is obtuse, its measure m satisfies 90 < m < 180. Halving every part keeps the order:
so each half is acute. It is never right, because that would need m = 180.
If the original is straight, each half is 180^\circ \div 2 = 90^\circ, so the halves are right angles.
Doubling works the other way and is less tidy. An acute angle has 0 < m < 90, so its double has 0 < 2m < 180, which runs across three kinds. For example 2 \times 30^\circ = 60^\circ is acute, 2 \times 45^\circ = 90^\circ is right, and 2 \times 70^\circ = 140^\circ is obtuse. Knowing only that an angle is acute tells you nothing about the kind of its double, while knowing that an angle is obtuse tells you exactly the kind of its half.
The angles of a triangle
Two angles of a triangle measure 35^\circ and 47^\circ. Find the third and name its kind. Then explain why no triangle can have two obtuse angles.
SolutionThe three angles of a triangle add to 180^\circ (the triangle angle sum), so the third is
which is obtuse. A triangle with an obtuse angle is called an obtuse triangle.
Now suppose a triangle had two obtuse angles. Each would be more than 90^\circ, so those two alone would add to more than 90^\circ + 90^\circ = 180^\circ, leaving nothing, less than nothing in fact, for the third angle. That is impossible, so a triangle has at most one obtuse angle. The same argument shows it has at most one right angle, and that a triangle with a right or obtuse angle has two acute ones. This is the fact that makes sorting triangles by their largest angle work.
Boundaries
The angle between two rays stops at 180^\circ. Two rays from one point make two turns, one each way round, and the angle named by the rays is the smaller one. That is why four names cover every such angle. A turn of more than 180^\circ and less than 360^\circ does exist, and is called a reflex angle; it appears as the inside angle of a concave polygon and in rotation, where a turn can go all the way round and past. It is not one of the four kinds here.
Zero is not acute. Two arms lying on top of each other make an angle of 0^\circ, which is outside every range above. Acute starts just above zero.
Right means exactly 90^\circ. An angle of 89.9^\circ is acute and one of 90.1^\circ is obtuse, though neither can be told from a right angle by eye. A diagram shows a right angle only where it is marked with a square or where the text says so. A corner that merely looks square is not evidence.
The kind does not depend on how the angle is drawn. Turning an angle round, flipping it over or lengthening its arms leaves its measure alone, and so leaves its kind alone. An acute angle drawn with its opening pointing down is still acute.
The landmarks do not depend on degrees. Measured in radians, a right angle is \displaystyle \frac{\pi}{2} and a straight angle is \pi. The numbers change; the four kinds and where the fences fall do not.
Where this goes next
The two landmarks each start a family of ideas. Two angles that fill a right angle between them are complementary, and two that fill a straight angle are supplementary; when those two sit side by side along a line they are a linear pair, which is the matched pair from the Results. Two lines that meet in a right angle are perpendicular, and one right angle at a crossing forces all four to be right.
Triangles are sorted by their angles into acute, right and obtuse triangles, and the right triangle goes on to carry the Pythagorean theorem and trigonometry.
Exercises
Naming the kind
Name the kind of an angle with each measure.
- 72^\circ
- 90^\circ
- 108^\circ
- 180^\circ
- 89.9^\circ
- 1^\circ
Angles along a line
- A ray leaves a point on a line and makes an angle of 64^\circ with the line on one side. Find the angle on the other side and name the kind of both.
- A ray leaves a point on a line and makes an angle of 90^\circ with the line on one side. Find the angle on the other side and name its kind.
- Two rays leave the same point on a line, on the same side, and split the straight angle into three angles. Two of them measure 40^\circ and 75^\circ. Find the third and name the kind of all three.
- Three angles fill a straight angle. Two of them measure 30^\circ and 45^\circ. Find the third and name its kind.
The hands of a clock
Find the smaller angle between the hands at each time, on the hour, and name its kind.
- Four o’clock.
- Nine o’clock.
- One o’clock.
- Seven o’clock.
Using algebra
- An angle measuring (2x + 10)^\circ is a right angle. Find x.
- An angle measuring (5x - 20)^\circ is a straight angle. Find x.
- An angle measuring (4x + 6)^\circ is acute. Find every value of x that is possible.
- Two angles fill a straight angle together. They measure (x + 30)^\circ and (2x)^\circ. Find x, then name the kind of each angle.
- An angle measuring (6x - 30)^\circ is obtuse. Find every value of x that is possible.
Reasoning
- Can two acute angles add up to an obtuse angle? Can they add up to a right angle? Explain, with an example for each answer that is yes.
- Points A, O and B lie on a line with O between A and B, and X is a point not on the line. Show that \angle XOA is acute exactly when \angle XOB is obtuse.
- Explain why two obtuse angles always add to more than a straight angle.
- Explain why half of any angle is acute, unless the angle is straight. What is half of a straight angle?
- A triangle has a right angle. Explain why its other two angles must both be acute.
Answers
- Acute. 0^\circ < 72^\circ < 90^\circ.
- Right. It is exactly 90^\circ.
- Obtuse. 90^\circ < 108^\circ < 180^\circ.
- Straight. It is exactly 180^\circ.
- Acute. 89.9^\circ is less than 90^\circ, however close.
- Acute. 1^\circ is more than 0^\circ and less than 90^\circ.
- 116^\circ; the 64^\circ angle is acute and the 116^\circ angle is obtuse. The two fill a straight angle, so the other is 180^\circ - 64^\circ = 116^\circ.
- 90^\circ, a right angle. 180^\circ - 90^\circ = 90^\circ, so both angles are right.
- 65^\circ, and all three are acute. The three fill 180^\circ, so the third is 180^\circ - 40^\circ - 75^\circ = 65^\circ, and 40, 75 and 65 are each less than 90.
- 105^\circ, obtuse. 180^\circ - 30^\circ - 45^\circ = 105^\circ, which is between 90^\circ and 180^\circ.
- 120^\circ, obtuse. Four gaps of 30^\circ: 4 \times 30^\circ = 120^\circ.
- 90^\circ, right. Counting forward gives 9 \times 30^\circ = 270^\circ, the larger way round, so the smaller angle is 360^\circ - 270^\circ = 90^\circ.
- 30^\circ, acute. One gap: 1 \times 30^\circ = 30^\circ.
- 150^\circ, obtuse. 7 \times 30^\circ = 210^\circ is the larger way round, so the smaller angle is 360^\circ - 210^\circ = 150^\circ.
- x = 40. A right angle is 90^\circ: 2x + 10 = 90, so 2x = 80.
- x = 40. A straight angle is 180^\circ: 5x - 20 = 180, so 5x = 200.
- -1.5 < x < 21. Acute means 0 < 4x + 6 < 90. Subtract 6: -6 < 4x < 84. Divide by 4.
- x = 50; the angles are 80^\circ, acute, and 100^\circ, obtuse. They add to 180: (x + 30) + 2x = 180, so 3x = 150. Then 50 + 30 = 80 and 2(50) = 100.
- 20 < x < 35. Obtuse means 90 < 6x - 30 < 180. Add 30: 120 < 6x < 210. Divide by 6.
- Yes to both. 50^\circ + 60^\circ = 110^\circ is obtuse, and 40^\circ + 50^\circ = 90^\circ is right. Two acute angles are each less than 90^\circ, so their sum is less than 180^\circ, but it can land on either side of 90^\circ or on it.
- \overrightarrow{OA} and \overrightarrow{OB} point in opposite directions, so \angle AOB is straight and m\angle XOA + m\angle XOB = 180^\circ. If m\angle XOA < 90^\circ then m\angle XOB = 180^\circ - m\angle XOA > 90^\circ, and it is less than 180^\circ because X is off the line, so \angle XOB is obtuse. Going the other way, if m\angle XOB > 90^\circ then m\angle XOA = 180^\circ - m\angle XOB < 90^\circ, and it is more than 0^\circ because X is off the line, so \angle XOA is acute.
- Each obtuse angle is more than 90^\circ, so the two together are more than 90^\circ + 90^\circ = 180^\circ.
- An angle that is not straight measures less than 180^\circ, so its half measures less than 90^\circ, and more than 0^\circ, so the half is acute. Half of a straight angle is exactly 90^\circ, a right angle.
- The three angles add to 180^\circ and the right angle uses 90^\circ, so the other two add to 90^\circ. Each is more than 0^\circ, so each is less than 90^\circ, which makes both acute.