Grade 4 · 4.MD.C.5 · G.CO.A.1

Angles and how to name them

Several angles can share one corner of a figure. How do you say which one you mean, so that nobody has to guess?

Unlocks

An angle is an amount of turnGrade 4 Angle types and pairsGrades 4 to 8 Measuring anglesGrades 4 to 8

Where you are 8 in Points, lines and planes

Points, lines and planes Points, lines, rays and segments Angle types and pairs8 Lines and planes in space ▾ Show all 8 (3 more)

When you reach for it

An angle is two rays that start from the same point. The point is the vertex, and the two rays are the sides. What makes an angle worth studying is the opening between the sides: how far one ray has to turn, the shorter way round, to lie along the other. An angle as an amount of turn and measuring angles give that opening a number of degrees. Before a turn can be measured, or two of them compared, you have to say which angle you mean. One angle drawn on its own needs no name: you can point at it.

Pointing stops working once a figure has more than one angle. A triangle has three corners; two crossing lines make four angles at one point. To write “this angle is 40^\circ” or “these two angles are equal”, you need a name that picks out exactly one angle and that any reader can trace back to the figure.

The procedure

There are three ways to name an angle. All three are in Figure 1.

Figure 1One angle, with its vertex at B and its sides through A and C.

By its vertex alone: \angle B. The symbol \angle is read “angle”, and the letter is the vertex. This is allowed only because exactly one angle in Figure 1 has its vertex at B, so the letter already picks it out.

By three points: \angle ABC. Take a point on one side, then the vertex, then a point on the other side. The middle letter is always the vertex; the outer two tell you which rays are the sides, \overrightarrow{BA} and \overrightarrow{BC}. This name works in every figure, however crowded, and it is the one to use when in doubt.

By a number or a small letter written in the opening: \angle 1. The figure marks the angle with an arc and a label, and the label becomes the name. It is quick, but it means nothing without the figure beside it.

So the angle in Figure 1 has four names: \angle B, \angle ABC, \angle CBA and \angle 1.

Why is the vertex alone not enough in general? Look at Figure 2.

Figure 2Three rays from O make three angles, so the letter O alone cannot say which one is meant.

Three rays start from O, and every pair of them makes an angle: the one marked 1, the one marked 2, and the larger one that contains both. “\angle O” could be any of them.

Three letters fix this, because an angle is decided by its two sides, and the three letters record exactly those two sides. The middle letter says where the rays start, and each outer letter is one other point on one ray, which is all it takes to fix a ray. So \angle AOB has sides \overrightarrow{OA} and \overrightarrow{OB}, and that is \angle 1 and nothing else. Two consequences follow.

The outer letters can be swapped. \angle AOB and \angle BOA have the same vertex and the same two sides, so they are the same angle.

The middle letter cannot move. \angle OAB would have its vertex at A, a different corner altogether. In Figure 2 there is no angle at A at all.

To read a three-letter name back into a figure, put your finger on the middle letter and follow the rays toward the other two. The opening between them is the angle.

Every pair of different rays counts as one angle, straight angles included. Two rays pointing in opposite directions still make an angle, the straight angle of 180^\circ, named like any other. So any two different rays from a point make exactly one angle, and that is how angles are counted in this lesson unless a question says otherwise.

The measure of an angle is written with an m in front: m\angle AOB = 45^\circ reads “the measure of angle A O B is 45 degrees”. The name \angle AOB is the figure; m\angle AOB is a number, the subject of measuring angles.

Worked examples

Example 1

Naming every angle of a triangle

Name each angle of triangle PQR in Figure 3 in three ways.

Figure 3A triangle has exactly one angle at each corner.

Solution

Each corner has exactly one angle, so a single letter is enough for each. For the three-letter names, the vertex goes in the middle and the other two corners go on the outside, because the two sides at each corner run to the other two corners.

At P the sides run to Q and to R: \angle P, \angle QPR, \angle RPQ.

At Q the sides run to P and to R: \angle Q, \angle PQR, \angle RQP.

At R the sides run to P and to Q: \angle R, \angle PRQ, \angle QRP.

Example 2

A crowded vertex

In Figure 2, give a three-letter name for \angle 1, for \angle 2, and for the angle between \overrightarrow{OA} and \overrightarrow{OC}. Explain why none of them can be called \angle O.

Solution

Each angle has its vertex at O, so O goes in the middle, and its two sides give the outer letters. \angle 1 has sides through A and B, so it is \angle AOB (or \angle BOA). \angle 2 has sides through B and C, so it is \angle BOC (or \angle COB). The angle between the outer rays has sides through A and C, so it is \angle AOC (or \angle COA).

All three have O as their vertex, so \angle O fits each of them equally well and picks out none.

Example 3

Many names for one angle

In Figure 4, A and D lie on one side of the angle at B, and C and E lie on the other. List every three-letter name for this angle. How many are there?

Figure 4Two named points on each side. Any one of them can stand for its side.

Solution

\overrightarrow{BA} and \overrightarrow{BD} are the same ray, since D lies on \overrightarrow{BA} beyond A. In the same way \overrightarrow{BC} and \overrightarrow{BE} are the same ray. So either A or D can stand for one side, and either C or E for the other.

A three-letter name is one choice for each side with the vertex between them, written in either order. That is 2 choices, times 2 choices, times 2 orders:

2 \times 2 \times 2 = 8

The eight names are

\angle ABC,\ \angle ABE,\ \angle DBC,\ \angle DBE

\angle CBA,\ \angle EBA,\ \angle CBD,\ \angle EBD

and since B is the vertex of only one angle in the figure, \angle B names it as well. Being names for one angle, they all carry one measure.

Example 4

The four angles at a crossing

Two lines cross at E, as in Figure 5. Name the four angles at E. What does \angle AEB name?

Figure 5Two lines crossing at E, with a named point on each of the four rays.

Solution

Four rays start at E: \overrightarrow{EA}, \overrightarrow{EC}, \overrightarrow{EB} and \overrightarrow{ED}, in that order going round. Each of the four angles lies between two neighboring rays, so its name uses the two points on those rays with E in the middle:

\angle AEC,\quad \angle CEB,\quad \angle BED,\quad \angle DEA

None of them can be called \angle E, since all four share that vertex.

\overrightarrow{EA} and \overrightarrow{EB} point in opposite directions along one line, so \angle AEB is a straight angle of 180^\circ: a proper name, but not for one of the four angles between neighboring rays, and silent about which side of line AB is meant. Which of the four angles are equal is the subject of vertical angles.

Example 5

One extra segment

In Figure 6, D lies on \overline{BC} and a segment joins A to D. Which corners can still be named with a single letter? How many angles in the figure can be named with three letters, not counting straight angles?

Figure 6One extra segment crowds two of the corners.

Solution

Work corner by corner, and at each one count the rays that start there.

At A three segments meet, going to B, D and C, and every pair makes an angle: \angle BAD, \angle DAC, \angle BAC. That is 3, so \angle A is not allowed.

At B only two sides meet, so there is 1 angle and \angle B is allowed. It is \angle ABC, or equally \angle ABD, since D lies on \overrightarrow{BC}. In the same way C has 1 angle, \angle ACB or \angle ACD, and \angle C is allowed.

At D three rays start, toward B, A and C. They make \angle ADB and \angle ADC, and also \angle BDC, a straight angle because B and C are in opposite directions from D. Not counting it, that is 2 angles, and \angle D is not allowed.

Only B and C can be named with one letter, and the count is

3 + 1 + 1 + 2 = 7

angles, with the straight angle \angle BDC as an eighth if you count it.

Example 6

Counting the angles at a point

Four rays start at O, as in Figure 7. How many angles can be named with three letters? How many would n rays give?

Figure 7Four rays from one point. How many angles are there?

Solution

Every angle at O is made by two of the rays, and every two of the rays make one angle. So counting angles means counting pairs of rays, in order so that nothing is missed or counted twice. Pair \overrightarrow{OA} with each ray after it: \angle AOB, \angle AOC, \angle AOD, which is 3. Pair \overrightarrow{OB} with each ray after it (its pair with \overrightarrow{OA} is already counted): \angle BOC, \angle BOD, which is 2. Then \overrightarrow{OC} with the one ray left: \angle COD, which is 1.

3 + 2 + 1 = 6

With n rays, each ray pairs with n - 1 others, which gives n(n - 1), but that counts every pair twice, once from each end. So n rays give

\frac{n(n - 1)}{2}

angles. For n = 4 this is \displaystyle \frac{4 \times 3}{2} = 6, as counted. The formula relies on the convention above: two opposite rays make a straight angle, which still counts as one. A count that leaves straight angles out must subtract them.

Boundaries

A single letter needs a lonely vertex. \angle B is a name only when exactly one angle in the figure has its vertex at B. Add one ray at B and the name stops working.

The outer points must lie on two different sides. If both outer letters lie on the same ray from the vertex, the two “sides” are one ray and there is no opening between them. At most that is a zero angle of 0^\circ, never one of the angles counted in a figure.

Three letters name the smaller opening. Two rays from a point split the whole turn into two openings, and by convention \angle AOB means the one of at most 180^\circ. The larger one is a reflex angle, and when you need it you say so in words: “the reflex angle AOB”.

A straight angle’s name does not choose a side. When B lies on line AC between A and C, \angle ABC is a straight angle of 180^\circ and could be either half of the plane. The measure is the same; anything that depends on the side needs more words.

Same measure is not same angle. Different names for one angle (like \angle ABC and \angle DBE in Example 3) always give the same measure. Two different angles with the same measure are congruent, not the same angle, and they keep different names.

Where this goes next

Almost every angle fact is a statement about named angles, and the names often carry the idea. Angle types and pairs puts the names and the measures to work together. Adjacent angles share a vertex letter and one outer letter, as \angle AOB and \angle BOC do, and the angle addition postulate says their measures add to m\angle AOC. An angle bisector \overrightarrow{BD} splits \angle ABC into \angle ABD and \angle DBC.

When a line crosses two others, eight angles appear at once, and they are usually numbered \angle 1 to \angle 8, because the relationships in parallel lines and transversals are about positions rather than points. And in triangles, \angle A, \angle B and \angle C are the standard names for the corners of triangle ABC.

Exercises

One angle, several names

Write four different names for \angle 1 in Figure 1.
In \angle PQR, which point is the vertex, and which two rays are the sides?
True or false: \angle XYZ and \angle ZYX name the same angle. Explain.
True or false: \angle XYZ and \angle YXZ name the same angle. Explain.

A crowded figure

Use Figure 8 for Exercises 5 to 9.

Figure 8A four-sided figure with one diagonal, for Exercises 5 to 9.

Explain why \angle A is not a good name for any angle in Figure 8.
Name the three angles with vertex A.
Which corners of Figure 8 can be named with a single letter?
How many angles in Figure 8 can be named with three letters?
Give both three-letter names for the angle at D.

Counting names and angles

In Exercises 11 to 14, as in the lesson, every pair of different rays from a point counts as one angle, straight angles included.

One side of an angle at B passes through the named points A and D, and the other side through C, E and F. How many three-letter names does the angle have?
Five different rays start from one point. How many angles can be named with three letters?
Ten different rays start from one point. How many angles can be named with three letters?
Some different rays start from one point, and 28 angles can be named with three letters. How many rays are there?
Two lines cross at E. A and B are on one line and C and D on the other, each pair on opposite sides of E. How many angles at E can be named with three letters, counting straight angles?

Names and measures

m\angle ABC = 35^\circ. What is m\angle CBA?
In Figure 4, m\angle ABC = 52^\circ. Find m\angle DBE.
In Figure 6, m\angle ABC = 64^\circ. Find m\angle ABD, and say why no calculation is needed.

Reasoning

Using Figure 3, explain why the vertex has to be the middle letter of a three-letter name.
A, B and C lie on one line, with B between A and C. What kind of angle is \angle ABC, and what does the name leave out?
A and C are two different points on the same ray from B, neither of them B. Does \angle ABC name an angle? Explain.
Two rays from O pass through A and B, and the opening between them on one side measures 100^\circ. What is m\angle AOB, and how large is the opening on the other side? How would you refer to that other opening?
Explain why n different rays from one point give exactly \displaystyle \frac{n(n - 1)}{2} angles that can be named with three letters, counting straight angles.

Answers

\angle B, \angle ABC, \angle CBA and \angle 1. B is the vertex of only one angle, and the outer letters can go either way.
The vertex is Q, the middle letter. The sides are \overrightarrow{QP} and \overrightarrow{QR}.
True. Both have vertex Y and sides \overrightarrow{YX} and \overrightarrow{YZ}; only the order of the outer letters differs.
False. \angle XYZ has its vertex at Y and \angle YXZ has its vertex at X, so they are angles at different corners.
Three segments meet at A, going to B, C and D, so three angles have their vertex at A and the letter alone cannot say which.
\angle DAC, \angle CAB and \angle DAB (each can also be written with its outer letters swapped).
B and D. Only two sides meet at each, so each is the vertex of exactly one angle. At A and C the diagonal adds a third ray.
8 angles. At A there are 3 (\angle DAC, \angle CAB, \angle DAB), at C there are 3 (\angle ACB, \angle BCD, \angle ACD), and at B and D one each: 3 + 3 + 1 + 1 = 8.
\angle ADC and \angle CDA. The sides at D run to A and to C.
12 names. There are 2 choices for the first side, 3 for the second, and 2 orders: 2 \times 3 \times 2 = 12.
10 angles. Each angle is one pair of rays, and \displaystyle \frac{5 \times 4}{2} = 10.
45 angles. \displaystyle \frac{10 \times 9}{2} = 45.
8 rays. \displaystyle \frac{n(n - 1)}{2} = 28 gives n^2 - n - 56 = 0, so (n - 8)(n + 7) = 0 and n = 8 or n = -7. A number of rays cannot be negative, so n = 8. Check: \displaystyle \frac{8 \times 7}{2} = 28.
6 angles. There are four rays at E, and \displaystyle \frac{4 \times 3}{2} = 6: the four angles \angle AEC, \angle CEB, \angle BED, \angle DEA, and the two straight angles \angle AEB and \angle CED.
m\angle CBA = 35^\circ. \angle CBA and \angle ABC are the same angle.
m\angle DBE = 52^\circ. D is on \overrightarrow{BA} and E is on \overrightarrow{BC}, so \angle DBE is the same angle as \angle ABC.
m\angle ABD = 64^\circ. D lies on \overrightarrow{BC}, so \angle ABD and \angle ABC have the same vertex and sides: one angle with two names.
The three letters P, Q, R can be arranged to put any corner of the triangle in the middle: \angle QPR, \angle PQR and \angle PRQ are three different angles. If the vertex could be any of the letters, one name would fit all three corners. Fixing it in the middle makes each name point to one corner.
A straight angle, measuring 180^\circ, since \overrightarrow{BA} and \overrightarrow{BC} point in opposite directions. The name does not say which side of the line is meant.
No. \overrightarrow{BA} and \overrightarrow{BC} are the same ray, so the two “sides” coincide and there is no opening between them; at most it is a zero angle of 0^\circ. The name fails because its outer letters do not pick out two different sides.
m\angle AOB = 100^\circ, because a three-letter name means the opening of at most 180^\circ. The other opening is 360^\circ - 100^\circ = 260^\circ, called in words the reflex angle AOB.
Each pair of different rays gives exactly one angle (a straight angle if they are opposite), and each angle comes from exactly one pair, so the angles match the pairs of rays. Each of the n rays pairs with the n - 1 others, giving n(n - 1), but this counts every pair twice, once from each of its rays. Dividing by 2 gives \displaystyle \frac{n(n - 1)}{2}.