How to Measure and Name Angles

Geometry starts with the simplest objects. A single point. Two points joined make a segment. Extend it in one direction and you get a ray. Extend it in both directions and you get a line. Now let two rays start from the same point. The opening between them is an angle. In this lesson you will learn to name angles and to measure how far they open, in degrees.

Key idea

Angles, names, and degrees. An angle is the opening between two rays that start at the same point. That shared point is the vertex and the two rays are the sides. Name an angle with three letters, always keeping the vertex in the middle, so

∠ A B C

is the angle at

B

. The order of the outer two letters is free, so

∠ A B C = ∠ C B A

, and when only one angle sits at a point you may just write

∠ B

. We measure how far an angle opens in degrees. A full turn is

36 0^{\circ}

, a straight line is half of that at

18 0^{\circ}

, and a square corner, called a right angle, is

9 0^{\circ}

. A protractor reads the degrees directly. We sort angles by how far they open. An acute angle is below

9 0^{\circ}

, right is exactly

9 0^{\circ}

, obtuse is between

9 0^{\circ}

and

18 0^{\circ}

, straight is exactly

18 0^{\circ}

, and reflex is more than

18 0^{\circ}

, measured the long way around.

Problem

In the diagram above,

∠ A B C

is a right angle (the small square at

B

marks it), and ray

B X

splits it into two parts. The diagram gives

∠ X B C = 2 8^{\circ}

. Find

∠ A B X

. Give the number of degrees.

Show a hint

The little square tells you $∠ A B C$ is a right angle, so the whole thing measures $9 0^{\circ}$ .
Ray $B X$ cuts that right angle into $∠ A B X$ and $∠ X B C$ , and those two parts add up to the full $9 0^{\circ}$ . You know one part already.

Show the full solution

The small square means

∠ A B C

is a right angle, so the two pieces ray

B X

makes must fill

9 0^{\circ}

. One piece is given, so the other is what's left over.

∠ A B X = 9 0^{\circ} - 2 8^{\circ} = 62

Problem

In the diagram above,

A

O

B

lie on a straight line and ray

O C

stands on it. The diagram gives

∠ A O C = 12 4^{\circ}

. Find

∠ B O C

. Give the number of degrees.

Show a hint

$∠ A O C$ and $∠ B O C$ sit side by side along the straight line $A B$ , so together they form a straight angle. How many degrees is a straight angle?
A straight angle is $18 0^{\circ}$ . Subtract the part you already know from $180$ to get the part you want.

Show the full solution

∠ A O C

and

∠ B O C

sit side by side along the straight line

A B

, so together they make a straight angle of

18 0^{\circ}

∠ B O C = 180 - 124 = 56

Problem

In the diagram above, ray

O B

lies inside

∠ A O C

, and

∠ A O B = 8 8^{\circ}

. The whole angle

∠ A O C = 14 7^{\circ}

. What is

∠ B O C

? Give the number of degrees.

Show a hint

Ray $O B$ cuts the big angle into two pieces, and those two pieces add up to the whole $∠ A O C$ .
You know the whole angle and one piece, so $∠ B O C$ is what is left over. Subtract.

Show the full solution

The two parts

∠ A O B

and

∠ B O C

add up to the whole angle

∠ A O C

. So

∠ B O C

is the whole minus the part you already know.

∠ B O C = 14 7^{\circ} - 8 8^{\circ} = 59

Problem

In the diagram above,

∠ A O C

is a right angle and ray

O D

splits it. Given that

∠ A O D = 3 7^{\circ}

, find

∠ D O C

. Give the number of degrees.

Show a hint

A right angle measures $9 0^{\circ}$ , and ray $O D$ breaks it into the two pieces $∠ A O D$ and $∠ D O C$ .
Those two pieces together make the whole right angle, so $∠ A O D + ∠ D O C = 9 0^{\circ}$ . Subtract to find the missing piece.

Show the full solution

Ray

O D

splits the right angle

∠ A O C

into two parts that add to

9 0^{\circ}

. The two angles are complementary, so

∠ D O C = 90 - 37 = 53 .

Problem

In the diagram above, two rays make an angle of

11 0^{\circ}

, and the question mark marks the reflex angle, the big way around the outside. What is the reflex angle? Give the number of degrees.

Show a hint

A full turn all the way around a point is $36 0^{\circ}$ . The reflex angle and the $11 0^{\circ}$ angle together sweep out that whole turn.
So the reflex angle is what is left after you take the $11 0^{\circ}$ away from $36 0^{\circ}$ .

Show the full solution

The

11 0^{\circ}

angle and the reflex angle together sweep the full turn around the vertex, which is

36 0^{\circ}

36 0^{\circ} - 11 0^{\circ} = 250

An angle and the reflex angle beside it always pair up to

36 0^{\circ}

, so either one hands you the other.

Problem

In the diagram above, ray

O Q

bisects

∠ P O R

, so the equal tick marks show the two halves are equal. Given that one half

∠ P O Q = 4 1^{\circ}

, what is the whole

∠ P O R

? Give the number of degrees.

Show a hint

A bisector cuts an angle into two equal halves, so $∠ P O R$ is made of two copies of $∠ P O Q$ .
The whole is twice one half. Double $4 1^{\circ}$ .

Show the full solution

A bisector splits an angle into two equal halves, so the whole is twice one half.

∠ P O R = 2 \times 4 1^{\circ} = 82^{\circ}

Problem

In the diagram above, two straight lines cross at a point, forming four angles. One of them measures

7 4^{\circ}

, and the angle marked with the question mark sits directly across from it. What is the marked angle? Give the number of degrees.

Show a hint

The two marked angles are directly across the crossing point from each other. What is that pair of angles called?
Vertical angles, the pair sitting opposite each other where two lines cross, are always equal. So the marked angle matches the $7 4^{\circ}$ one.

Show the full solution

The marked angle sits directly across the crossing point from the

7 4^{\circ}

angle, so the two are vertical angles and have the same measure. The marked angle is

74

degrees. Vertical angles match because each one shares a straight line with the same neighbor, so both come out as

18 0^{\circ}

minus that neighbor.

Problem

In the diagram above, three rays leave a single point, cutting the whole way around into three angles. Two of them measure

13 7^{\circ}

and

9 5^{\circ}

. Find the third angle, the one marked with the question mark. Give the number of degrees.

Show a hint

Going all the way around a point is one full turn, and a full turn is $36 0^{\circ}$ . The three angles have to add up to that.
So the missing angle is what's left after you take the two known ones away from $36 0^{\circ}$ .

Show the full solution

The three angles wrap all the way around the point, so they add to a full turn of

36 0^{\circ}

. Subtract the two you know.

360 - 137 - 95 = 128

Problem

In the diagram above, a ray splits the right angle

∠ A B C

into two parts whose measures are in the ratio

1 : 5

. Find the smaller part, the one marked with the question mark. Give the number of degrees.

Show a hint

A right angle is $9 0^{\circ}$ , and the two parts together make up that whole $9 0^{\circ}$ .
A ratio of $1 : 5$ means the angle is cut into $1 + 5 = 6$ equal shares. The smaller part is just one of those shares.

Show the full solution

The ratio

1 : 5

splits the right angle into

1 + 5 = 6

equal shares. Since a right angle is

9 0^{\circ}

, each share is

\frac{9 0^{\circ}}{6} = 1 5^{\circ} .

The smaller part is one share, so it measures

15

degrees.

Problem

The clock in the diagram above shows 5:00. Find the measure of the (non-reflex) angle between the hour hand and the minute hand. Give the number of degrees.

Show a hint

The 12 hour marks split the full $36 0^{\circ}$ evenly. How many degrees is one gap between neighboring marks?
Count how many of those gaps sit between the two hands at 5:00, then add up that many equal pieces.

Show the full solution

The 12 marks divide the full circle evenly, so each gap is

\frac{36 0^{\circ}}{12} = 3 0^{\circ}

. At 5:00 the minute hand points at 12 and the hour hand points at 5, which is 5 gaps away. So the angle is

5 \times 3 0^{\circ} = 150

degrees.

Problem

In the diagram above, a ray stands on a straight line, and the two angles it forms are labeled

x^{\circ}

and

(x + 10)^{\circ}

. Find

x

. Give the number of degrees (the value of

x

Show a hint

The ray sits on a straight line, so the two angles together open up all the way across it. What does a straight angle measure?
The two labeled angles are supplementary, so they add to $18 0^{\circ}$ . Set up $x + (x + 10) = 180$ and solve for $x$ .

Show the full solution

A ray standing on a straight line makes two supplementary angles, so they add to

18 0^{\circ}

. Adding the labels gives

x + (x + 10) = 180,

which simplifies to

2 x + 10 = 180

. Then

2 x = 170

, so

x = 85

Problem

In the diagram above, two rays split the straight angle into three angles whose measures are in the ratio

2 : 3 : 4

. Find the largest of the three, the one marked with the question mark. Give the number of degrees.

Show a hint

A straight angle measures $18 0^{\circ}$ , and the three parts together fill it. Think of the ratio $2 : 3 : 4$ as splitting that $18 0^{\circ}$ into equal shares.
Add the ratio numbers: $2 + 3 + 4 = 9$ . So $18 0^{\circ}$ is cut into $9$ equal shares. The largest angle takes $4$ of them.

Show the full solution

The three angles sit along a straight line, so they add to

18 0^{\circ}

. The ratio

2 : 3 : 4

cuts that into

2 + 3 + 4 = 9

equal shares, so each share is

\frac{18 0^{\circ}}{9} = 2 0^{\circ}

. The largest angle is

4

shares.

\frac{4}{9} \times 18 0^{\circ} = 4 \times 2 0^{\circ} = 80

Problem

In the diagram above, an angle

x^{\circ}

and its supplement

(180 - x)^{\circ}

sit side by side on a straight line. The angle is

1 5^{\circ}

more than twice its supplement. Find the angle. Give the number of degrees.

Show a hint

Two angles that sit together on a straight line are supplementary, so they add to $18 0^{\circ}$ . That is why the supplement is written $(180 - x)^{\circ}$ .
Turn the words into an equation. " $1 5^{\circ}$ more than twice its supplement" means $x = 2 (180 - x) + 15$ . Expand the right side and collect the $x$ terms.

Show the full solution

The two angles sit together on a straight line, so the supplement of

x^{\circ}

(180 - x)^{\circ}

. The angle is

1 5^{\circ}

more than twice that, so

x = 2 (180 - x) + 15.

Expanding gives

x = 360 - 2 x + 15

, so

3 x = 375

and

x = 125

Practice these ideas

Practice

In the diagram above,

∠ A B C

is a right angle and ray

B X

splits it into two smaller angles. Given that

∠ X B C = 3 4^{\circ}

, what is

∠ A B X

? Give the number of degrees.

Show the solution

The two pieces sit side by side and fill the right angle, so they add to

9 0^{\circ}

. That means

∠ A B X

is whatever is left after taking away

∠ X B C

∠ A B X = 9 0^{\circ} - 3 4^{\circ} = 56

Practice

In the diagram above, ray

O B

lies inside

∠ A O C

, splitting it into two smaller angles. The diagram shows

∠ A O B = 9 2^{\circ}

, and the whole

∠ A O C = 15 5^{\circ}

. What is

∠ B O C

? Give the number of degrees.

Show the solution

∠ A O B

and

∠ B O C

are the two pieces of

∠ A O C

, so the missing piece is the whole minus the part you know.

∠ B O C = 15 5^{\circ} - 9 2^{\circ} = 63

Practice

In the diagram above,

A O B

is a straight line and ray

O C

rises from point

O

. Given that

∠ A O C = 11 3^{\circ}

, what is

∠ B O C

? Give the number of degrees.

Show the solution

Angles on a straight line add up to

18 0^{\circ}

. Since

∠ A O C

and

∠ B O C

together form the straight line

A O B

∠ B O C = 18 0^{\circ} - 11 3^{\circ} = 67^{\circ} .

Practice

In the diagram above,

∠ A O C

is a right angle and ray

O D

falls inside it. Given that

∠ A O D = 2 9^{\circ}

, what is

∠ D O C

? Give the number of degrees.

Show the solution

The two smaller angles sit inside the right angle and add up to it, so

∠ A O D + ∠ D O C = 9 0^{\circ}

. Subtract the part you know.

∠ D O C = 9 0^{\circ} - 2 9^{\circ} = 61

Practice

In the diagram above, ray

O Q

bisects

∠ P O R

, so the two halves are equal. Given that

∠ P O Q = 3 8^{\circ}

, what is the whole

∠ P O R

? Give the number of degrees.

Show the solution

A bisector cuts an angle into two equal halves, so the whole is twice one half.

∠ P O R = 2 \times 3 8^{\circ} = 76

Practice

In the diagram above, two lines cross and one of the four angles measures

11 6^{\circ}

. The marked angle sits right beside it along the same straight line. What is the measure of the marked angle? Give the number of degrees.

Show the solution

The marked angle and the

11 6^{\circ}

angle sit side by side along a straight line, so they add to

18 0^{\circ}

18 0^{\circ} - 11 6^{\circ} = 64

Where two lines cross, every angle is either equal to a given one or supplementary to it.

Practice

In the diagram above, three rays from a single point split the space all the way around into three angles. Two of them measure

8 8^{\circ}

and

14 5^{\circ}

. What is the third angle? Give the number of degrees.

Show the solution

The three angles fill a complete turn around the point, so they add to

36 0^{\circ}

. Subtract the two you know.

360 - 88 - 145 = 127

Practice

In the diagram above, two rays open up to make an angle of

14 5^{\circ}

. The question mark marks the reflex angle, the one that sweeps all the way around the outside. What is the reflex angle? Give the number of degrees.

Show the solution

The two angles at the vertex fill one complete turn, so together they add to

36 0^{\circ}

. Subtract the part you know from the whole turn.

36 0^{\circ} - 14 5^{\circ} = 215^{\circ}

Practice

In the diagram above, ray

O P

stands on a straight line, splitting the straight angle into

x^{\circ}

and

2 x^{\circ}

. Find

x

. Give the number of degrees.

Show the solution

The two angles rest on a straight line, so together they form a straight angle of

18 0^{\circ}

. Adding the pieces gives

x + 2 x = 3 x = 180,

x = 60

Practice

In the diagram above, ray

B X

splits the right angle

∠ A B C

into two parts whose measures are in the ratio

2 : 7

. What is the measure of the smaller part? Give the number of degrees.

Show the solution

The ratio

2 : 7

cuts the right angle into

2 + 7 = 9

equal shares, so each share is

9 0^{\circ} \div 9 = 1 0^{\circ}

. The smaller part is

2

shares.

\frac{2}{9} \times 9 0^{\circ} = 20

Practice

In the diagram above, the angle

y^{\circ}

and its complement

(90 - y)^{\circ}

together fill a right angle. The angle exceeds its complement by

4 0^{\circ}

. What is

y

? Give the number of degrees.

Show the solution

The angle is

4 0^{\circ}

more than its complement, so

y = (90 - y) + 40.

The right side is

130 - y

, so

2 y = 130

and

y = 65

. The complement is then

2 5^{\circ}

, and

65 - 25 = 40

checks out.

Practice

The clock in the diagram above reads 3:00. What is the non-reflex angle between the hour hand and the minute hand? Give the number of degrees.

Show the solution

The clock face is a full turn of

36 0^{\circ}

divided into 12 equal hour steps, so each step is

36 0^{\circ} \div 12 = 3 0^{\circ}

. At 3:00 the hour hand points at 3 and the minute hand points at 12, which are 3 steps apart. So the angle is

3 \times 3 0^{\circ} = 90^{\circ}

Practice

In the diagram above, three rays from a single point split the full turn around that point into three angles whose measures are in the ratio

3 : 4 : 5

. What is the measure of the largest of these three angles? Give the number of degrees.

Show the solution

The three angles go all the way around the point, so they add to a full turn of

36 0^{\circ}

. The ratio

3 : 4 : 5

splits that turn into

3 + 4 + 5 = 12

equal shares, so one share is

\frac{36 0^{\circ}}{12} = 3 0^{\circ}

. The largest angle is

5

shares.

5 \times 3 0^{\circ} = \frac{5}{12} \times 36 0^{\circ} = 150

Measuring Angles

Practice these ideas