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Angles

What is an Angle?

Comparing Angles

Mcq
Which angle has the greater measure?



The measure of an angle depends on the opening between its rays. A wider opening means a greater angle measure. Angle B has a wider opening (\(120^\circ\)) compared to Angle A (\(30^\circ\)). Therefore, Angle B is greater.

Mcq
Which angle has the greater measure?



The measure of an angle depends on the opening between its rays. A wider opening means a greater angle measure. Angle B has a wider opening (\(100^\circ\)) compared to Angle A (\(30^\circ\)). Therefore, Angle B is greater.

Mcq
Which angle has the greater measure?



The measure of an angle depends on the opening between its rays. A wider opening means a greater angle measure. Angle A has a wider opening (\(170^\circ\)) compared to Angle B (\(80^\circ\)). Therefore, Angle A is greater.

Mcq
Which angle has the greater measure?



The measure of an angle depends on the opening between its rays. A wider opening means a greater angle measure. Angle A has a wider opening (\(60^\circ\)) compared to Angle B (\(30^\circ\)). Therefore, Angle A is greater.

Mcq
Which angle has the greater measure?



The measure of an angle depends on the opening between its rays. A wider opening means a greater angle measure. Angle A has a wider opening (\(90^\circ\)) compared to Angle B (\(30^\circ\)). Therefore, Angle A is greater.

Naming Angles with Three Points

Mcq
Which option correctly names the marked angle using three-point notation?




The marked angle has vertex \(D\), with points \(E\) and \(F\) on its sides. In three-point notation, the vertex is in the middle, so the correct name is \(\Angle{FDE}\).

Mcq
Which option correctly names the marked angle using three-point notation?




The marked angle has vertex \(F\), with points \(D\) and \(E\) on its sides. In three-point notation, the vertex is in the middle, so the correct name is \(\Angle{DFE}\).

Mcq
Which option correctly names the marked angle using three-point notation?




The marked angle has vertex \(E\), with points \(D\) and \(F\) on its sides. In three-point notation, the vertex is in the middle, so the correct name is \(\Angle{DEF}\).

Mcq
Which option correctly names the marked angle using three-point notation?





The marked angle has vertex \(A\), with points \(D\) and \(B\) on its sides. In three-point notation, the vertex is in the middle, so the correct name is \(\Angle{DAB}\).

Mcq
Which option correctly names the marked angle using three-point notation?





The marked angle has vertex \(F\), with points \(D\) and \(E\) on its sides. In three-point notation, the vertex is in the middle, so the correct name is \(\Angle{DFE}\).

Mcq
Which option correctly names the marked angle using three-point notation?





The marked angle has vertex \(R\), with points \(T\) and \(Z\) on its sides. In three-point notation, the vertex is in the middle, so the correct name is \(\Angle{ZRT}\).

Degrees

Dividing the Full Turn

Exercise

One-half of a full turn measures \(^\circ\).

$$\begin{aligned}\text{One-half of a full turn} &= \frac{1}{2} \times 360^\circ \\ &= 360^\circ \div 2 \\ &= 180^\circ\end{aligned}$$

Exercise

One-quarter of a full turn measures \(^\circ\).

$$\begin{aligned}\text{One-quarter of a full turn} &= \frac{1}{4} \times 360^\circ \\ &= 360^\circ \div 4 \\ &= 90^\circ\end{aligned}$$

Exercise

One-sixth of a full turn measures \(^\circ\).

$$\begin{aligned}\text{One-sixth of a full turn} &= \frac{1}{6} \times 360^\circ \\ &= 360^\circ \div 6 \\ &= 60^\circ\end{aligned}$$

Exercise

One-eighth of a full turn measures \(^\circ\).

$$\begin{aligned}\text{One-eighth of a full turn} &= \frac{1}{8} \times 360^\circ \\ &= 360^\circ \div 8 \\ &= 45^\circ\end{aligned}$$

Exercise

One-third of a full turn measures \(^\circ\).

$$\begin{aligned}\text{One-third of a full turn} &= \frac{1}{3} \times 360^\circ \\ &= 360^\circ \div 3 \\ &= 120^\circ\end{aligned}$$

Measuring and Drawing Angles with a Protractor

Measuring Angles

Exercise
The angle shown measures \(^\circ\).

To measure an angle with a protractor, place its center on the vertex and align one ray with the \(0^\circ\) mark. The other ray points to the angle’s measure on the protractor’s scale. Here, one ray aligns with \(0^\circ\), and the other points to \(50^\circ\), so the angle measures \(50^\circ\).

Exercise
The angle shown measures \(^\circ\).

To measure an angle with a protractor, place its center on the vertex and align one ray with the \(0^\circ\) mark. The other ray points to the angle’s measure on the protractor’s scale. Here, one ray aligns with \(0^\circ\), and the other points to \(30^\circ\), so the angle measures \(30^\circ\).

Exercise
The angle shown measures \(^\circ\).

To measure an angle with a protractor, place its center on the vertex and align one ray with the \(0^\circ\) mark. The other ray points to the angle’s measure on the protractor’s scale. Here, one ray aligns with \(0^\circ\), and the other points to \(100^\circ\), so the angle measures \(100^\circ\).

Exercise
The angle shown measures \(^\circ\).

To measure an angle with a protractor, place its center on the vertex and align one ray with the \(0^\circ\) mark. The other ray points to the angle’s measure on the protractor’s scale. Here, one ray aligns with \(0^\circ\), and the other points to \(90^\circ\), so the angle measures \(90^\circ\).

Exercise
The angle shown measures \(^\circ\).

To measure an angle with a protractor, place its center on the vertex and align one ray with the \(0^\circ\) mark. The other ray points to the angle’s measure on the protractor’s scale. Here, one ray aligns with \(0^\circ\), and the other points to \(120^\circ\), so the angle measures \(120^\circ\).

Exercise
The angle shown measures \(^\circ\).

To measure an angle with a protractor, place its center on the vertex and align one ray with the \(0^\circ\) mark. The other ray points to the angle’s measure on the protractor’s scale. Here, one ray aligns with \(0^\circ\), and the other points to \(115^\circ\), so the angle measures \(115^\circ\).

Exercise
The angle shown measures \(^\circ\).

To measure an angle with a protractor, place its center on the vertex and align one ray with the \(0^\circ\) mark. The other ray points to the angle’s measure on the protractor’s scale. Here, one ray aligns with \(0^\circ\), and the other points to \(45^\circ\), so the angle measures \(45^\circ\).

Measuring Angles

Mcq
Using a protractor, find the measure of the angle shown.





To measure an angle with a protractor, place its center on the vertex and align one ray with the \(0^\circ\) mark. The other ray points to the angle’s measure on the protractor’s scale.
Here, one ray aligns with \(0^\circ\), and the other points to \(50^\circ\), so the angle measures \(50^\circ\).

Mcq
Using a protractor, find the measure of the angle shown.





To measure an angle with a protractor, place its center on the vertex and align one ray with the \(0^\circ\) mark. The other ray points to the angle’s measure on the protractor’s scale.
Here, one ray aligns with \(0^\circ\), and the other points to \(30^\circ\), so the angle measures \(30^\circ\).

Mcq
Using a protractor, find the measure of the angle shown.





To measure an angle with a protractor, place its center on the vertex and align one ray with the \(0^\circ\) mark. The other ray points to the angle’s measure on the protractor’s scale.
Here, one ray aligns with \(0^\circ\), and the other points to \(100^\circ\), so the angle measures \(100^\circ\).

Mcq
Using a protractor, find the measure of the angle shown.





To measure an angle with a protractor, place its center on the vertex and align one ray with the \(0^\circ\) mark. The other ray points to the angle’s measure on the protractor’s scale.
Here, one ray aligns with \(0^\circ\), and the other points to \(90^\circ\), so the angle measures \(90^\circ\).

Mcq
Using a protractor, find the measure of the angle shown.





To measure an angle with a protractor, place its center on the vertex and align one ray with the \(0^\circ\) mark. The other ray points to the angle’s measure on the protractor’s scale.
Here, one ray aligns with \(0^\circ\), and the other points to \(120^\circ\), so the angle measures \(120^\circ\).

Constructing Angles

Exercise
Using a pencil, a ruler, and a protractor, draw an angle that measures \(90^\circ\).

To draw a \(90^\circ\) angle:
  1. Draw a ray using a ruler to create the first side of the angle.
  2. Place the protractor’s center on the endpoint of the ray (the vertex) and align the baseline with the ray at \(0^\circ\).
  3. Mark a point at \(90^\circ\) on the protractor’s scale.
  4. Remove the protractor and use the ruler to draw a second ray from the vertex through the marked point.
The resulting angle measures \(90^\circ\), as shown below.

Exercise
Using a pencil, a ruler, and a protractor, draw an angle that measures \(60^\circ\).

To draw a \(60^\circ\) angle:
  1. Draw a ray using a ruler to create the first side of the angle.
  2. Place the protractor’s center on the endpoint of the ray (the vertex) and align the baseline with the ray at \(0^\circ\).
  3. Mark a point at \(60^\circ\) on the protractor’s scale.
  4. Remove the protractor and use the ruler to draw a second ray from the vertex through the marked point.
The resulting angle measures \(60^\circ\), as shown below.

Exercise
Using a pencil, a ruler, and a protractor, draw an angle that measures \(120^\circ\).

To draw a \(120^\circ\) angle:
  1. Draw a ray using a ruler to create the first side of the angle.
  2. Place the protractor’s center on the endpoint of the ray (the vertex) and align the baseline with the ray at \(0^\circ\).
  3. Mark a point at \(120^\circ\) on the protractor’s scale.
  4. Remove the protractor and use the ruler to draw a second ray from the vertex through the marked point.
The resulting angle measures \(120^\circ\), as shown below.

Exercise
Using a pencil, a ruler, and a protractor, draw an angle that measures \(45^\circ\).

To draw a \(45^\circ\) angle:
  1. Draw a ray using a ruler to create the first side of the angle.
  2. Place the protractor’s center on the endpoint of the ray (the vertex) and align the baseline with the ray at \(0^\circ\).
  3. Mark a point at \(45^\circ\) on the protractor’s scale.
  4. Remove the protractor and use the ruler to draw a second ray from the vertex through the marked point.
The resulting angle measures \(45^\circ\), as shown below.

Classification of Angles

Identifying Angle Types by Measure

Mcq
What is the nature of the marked angle?
Choose one answer:





  • An acute angle measures less than 90 degrees.
  • The marked angle, measuring \(40^\circ\), is acute because it is less than \(90^\circ\).

Mcq
What is the nature of the marked angle?
Choose one answer:





  • An obtuse angle measures more than 90 degrees but less than 180 degrees.
  • The marked angle, measuring \(110^\circ\), is obtuse because it is between \(90^\circ\) and \(180^\circ\).

Mcq
What is the nature of the marked angle?
Choose one answer:





  • A right angle measures exactly 90 degrees.
  • The marked angle, measuring \(90^\circ\), is a right angle.

Mcq
What is the nature of the marked angle?
Choose one answer:





  • An acute angle measures less than 90 degrees.
  • The marked angle, measuring \(45^\circ\), is acute because it is less than \(90^\circ\).

Mcq
What is the nature of the marked angle?
Choose one answer:





  • An obtuse angle measures more than 90 degrees but less than 180 degrees.
  • The marked angle, measuring \(135^\circ\), is obtuse because it is between \(90^\circ\) and \(180^\circ\).

Identifying Angle Types

Mcq
Identify the type of the highlighted angle.
Choose one answer:





  • An acute angle measures less than \(90^\circ\).
  • The highlighted angle (\(\approx 40^\circ\)) is less open than a right angle .
  • Hence it is acute.

Mcq
Identify the type of the highlighted angle.
Choose one answer:





  • An obtuse angle measures between \(90^\circ\) and \(180^\circ\).
  • The highlighted angle (\(\approx160^\circ\)) is more open than a right angle but less than a straight angle .
  • Therefore it is obtuse.

Mcq
Identify the type of the highlighted angle.
Choose one answer:





  • A straight angle measures exactly \(180^\circ\).
  • The highlighted angle forms a line.
  • It is therefore straight.

Mcq
Identify the type of the highlighted angle.
Choose one answer:





  • An obtuse angle measures between \(90^\circ\) and \(180^\circ\).
  • The highlighted angle (\(\approx110^\circ\)) is more open than a right angle but less open than a straight angle .
  • Therefore it is obtuse.

Constructing Angle Types

Exercise
Using a pencil, a ruler, and a protractor, draw an acute angle.

To draw an acute angle, such as a \(40^\circ\) angle:
  1. Draw a ray using a ruler to create the first side of the angle.
  2. Place the protractor’s center on the endpoint of the ray (the vertex) and align the baseline with the ray at \(0^\circ\).
  3. Mark a point at \(40^\circ\) on the protractor’s scale (any angle less than \(90^\circ\) is acceptable).
  4. Remove the protractor and use the ruler to draw a second ray from the vertex through the marked point.
The resulting angle is acute, measuring less than \(90^\circ\), as shown below.

Exercise
Using a pencil, a ruler, and a protractor, draw an obtuse angle.

To draw an obtuse angle, such as a \(120^\circ\) angle:
  1. Draw a ray using a ruler to create the first side of the angle.
  2. Place the protractor’s center on the endpoint of the ray (the vertex) and align the baseline with the ray at \(0^\circ\).
  3. Mark a point at \(120^\circ\) on the protractor’s scale (any angle greater than \(90^\circ\) but less than \(180^\circ\) is acceptable).
  4. Remove the protractor and use the ruler to draw a second ray from the vertex through the marked point.
The resulting angle is obtuse, measuring greater than \(90^\circ\) but less than \(180^\circ\), as shown below.

Exercise
Using a pencil, a ruler, and a protractor, draw a right angle.

To draw a right angle, which measures \(90^\circ\):
  1. Draw a ray using a ruler to create the first side of the angle.
  2. Place the protractor’s center on the endpoint of the ray (the vertex) and align the baseline with the ray at \(0^\circ\).
  3. Mark a point at \(90^\circ\) on the protractor’s scale.
  4. Remove the protractor and use the ruler to draw a second ray from the vertex through the marked point.
The resulting angle is a right angle, measuring exactly \(90^\circ\), as shown below.

Angle Addition

Adding Angles

Exercise
Calculate the measure of \(\Angle{ABC}\).

\(\Angle{ABC}=\)\(^\circ\)

Using the angle addition postulate, \(\Angle{ABC}\) is the sum of the smaller angles \(\Angle{ABD}\) and \(\Angle{DBC}\):$$\begin{aligned}\Angle{ABC} &= \Angle{ABD} + \Angle{DBC} \\ &= 70^\circ + 50^\circ \\ &= 120^\circ\end{aligned}$$

Exercise
Calculate the measure of \(\Angle{FER}\).

\(\Angle{FER}=\)\(^\circ\)

Using the angle addition postulate, \(\Angle{FER}\) is the sum of the smaller angles \(\Angle{FED}\) and \(\Angle{DER}\):$$\begin{aligned}\Angle{FER} &= \Angle{FED} + \Angle{DER} \\ &= 30^\circ + 75^\circ \\ &= 105^\circ\end{aligned}$$

Exercise
Calculate the measure of \(\Angle{FER}\).

\(\Angle{FER}=\)\(^\circ\)

Using the angle addition postulate, \(\Angle{FER}\) is the sum of the smaller angles \(\Angle{FED}\) and \(\Angle{DER}\):$$\begin{aligned}\Angle{FER} &= \Angle{FED} + \Angle{DER} \\ &= 110^\circ + 130^\circ \\ &= 240^\circ\end{aligned}$$

Exercise
Calculate the measure of \(\Angle{FEM}\).

\(\Angle{FEM}=\)\(^\circ\)

Using the angle addition postulate, \(\Angle{FEM}\) is the sum of the smaller angles \(\Angle{FED}\), \(\Angle{DER}\), and \(\Angle{REM}\):$$\begin{aligned}\Angle{FEM} &= \Angle{FED} + \Angle{DER} + \Angle{REM} \\ &= 55^\circ + 45^\circ + 60^\circ \\ &= 160^\circ\end{aligned}$$

Exercise
Calculate the measure of \(\Angle{FXM}\).

\(\Angle{FXM}=\)\(^\circ\)

Using the angle addition postulate, \(\Angle{FXM}\) is the sum of the smaller angles \(\Angle{FXD}\), \(\Angle{DXR}\), and \(\Angle{RXM}\):$$\begin{aligned}\Angle{FXM} &= \Angle{FXD} + \Angle{DXR} + \Angle{RXM} \\ &= 30^\circ + 60^\circ + 50^\circ \\ &= 140^\circ\end{aligned}$$

Exercise
Calculate the measure of \(\Angle{MZP}\).

\(\Angle{MZP}=\)\(^\circ\)

Using the angle addition postulate, \(\Angle{MZP}\) is the sum of the smaller angles \(\Angle{MZR}\), \(\Angle{RZD}\), and \(\Angle{DZP}\):$$\begin{aligned}\Angle{MZP} &= \Angle{MZR} + \Angle{RZD} + \Angle{DZP} \\ &= 60^\circ + 40^\circ + 80^\circ \\ &= 180^\circ\end{aligned}$$

Subtracting Angles

Exercise
Calculate the measure of \(\Angle{CBM}\).

\(\Angle{CBM}=\)\(^\circ\)

Using the angle addition postulate, the larger angle is the sum of the smaller angles:$$\Angle{CBM} + \Angle{MBA} = \Angle{CBA}$$To find \(\Angle{CBM}\), subtract \(\Angle{MBA}\) from \(\Angle{CBA}\):$$\begin{aligned}\Angle{CBM} &= \Angle{CBA} - \Angle{MBA} \\ &= 140^\circ - 70^\circ \\ &= 70^\circ\end{aligned}$$

Exercise
Calculate the measure of \(\Angle{FDN}\).

\(\Angle{FDN}=\)\(^\circ\)

Using the angle addition postulate, the larger angle is the sum of the smaller angles:$$\Angle{FDN} + \Angle{NDE} = \Angle{FDE}$$To find \(\Angle{FDN}\), subtract \(\Angle{NDE}\) from \(\Angle{FDE}\):$$\begin{aligned}\Angle{FDN} &= \Angle{FDE} - \Angle{NDE} \\ &= 120^\circ - 50^\circ \\ &= 70^\circ\end{aligned}$$

Exercise
Calculate the measure of \(\Angle{IGJ}\).

\(\Angle{IGJ}=\)\(^\circ\)

Using the angle addition postulate, the larger angle is the sum of the smaller angles:$$\Angle{IGJ} + \Angle{JGH} = \Angle{IGH}$$To find \(\Angle{IGJ}\), subtract \(\Angle{JGH}\) from \(\Angle{IGH}\):$$\begin{aligned}\Angle{IGJ} &= \Angle{IGH} - \Angle{JGH} \\ &= 160^\circ - 30^\circ \\ &= 130^\circ\end{aligned}$$

Exercise
Calculate the measure of \(\Angle{DZP}\) by subtracting the known angles from the larger angle using the angle addition postulate.

\(\Angle{DZP}=\)\(^\circ\)

Using the angle addition postulate, the larger angle is the sum of the smaller angles:$$\Angle{MZR} + \Angle{RZD} + \Angle{DZP} = \Angle{MZP}$$To find \(\Angle{DZP}\), subtract \(\Angle{MZR}\) and \(\Angle{RZD}\) from \(\Angle{MZP}\):$$\begin{aligned}\Angle{DZP} &= \Angle{MZP} - \Angle{MZR} - \Angle{RZD} \\ &= 180^\circ - 60^\circ - 40^\circ \\ &= 80^\circ\end{aligned}$$

Exercise
Calculate the measure of \(\Angle{AOB}\) by subtracting the known angles from the larger angle using the angle addition postulate.

\(\Angle{AOB}=\)\(^\circ\)

Using the angle addition postulate, the larger angle is the sum of the smaller angles:$$\Angle{AOB} + \Angle{BOC} + \Angle{COD} = \Angle{AOD}$$To find \(\Angle{AOB}\), subtract \(\Angle{BOC}\) and \(\Angle{COD}\) from \(\Angle{AOD}\):$$\begin{aligned}\Angle{AOB} &= \Angle{AOD} - \Angle{BOC} - \Angle{COD} \\ &= 160^\circ - 60^\circ - 60^\circ \\ &= 40^\circ\end{aligned}$$

Angle Properties

Calculating an Unknown Angle in a Right Angle

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles in a right angle is equal to \(90^\circ\).$$\begin{aligned}x^\circ + 50^\circ &= 90^\circ \\ x^\circ &= 90^\circ - 50^\circ \quad (\text{subtract } 50^\circ) \\ &= 40^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles in a right angle is equal to \(90^\circ\).$$\begin{aligned}25^\circ + x^\circ &= 90^\circ \\ x^\circ &= 90^\circ - 25^\circ \quad (\text{subtract } 25^\circ) \\ &= 65^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles in a right angle is equal to \(90^\circ\).$$\begin{aligned}30^\circ + x^\circ &= 90^\circ \\ x^\circ &= 90^\circ - 30^\circ \quad (\text{subtract } 30^\circ) \\ &= 60^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles in a right angle is equal to \(90^\circ\). The two angles are equal (\(x^\circ\)).$$\begin{aligned}x^\circ + x^\circ &= 90^\circ \\ 2x^\circ &= 90^\circ \quad (\text{combine like terms}) \\ x^\circ &= 90^\circ \div 2 \quad (\text{divide by } 2) \\ &= 45^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles in a right angle is equal to \(90^\circ\). The three angles are equal (\(x^\circ\)).$$\begin{aligned}x^\circ + x^\circ + x^\circ &= 90^\circ \\ 3x^\circ &= 90^\circ \quad (\text{combine like terms}) \\ x^\circ &= 90^\circ \div 3 \quad (\text{divide by } 3) \\ &= 30^\circ\end{aligned}$$

Calculating an Unknown Angle in a Straight Angle

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles on a straight line is equal to \(180^\circ\).$$\begin{aligned}x^\circ + 80^\circ &= 180^\circ \\ x^\circ &= 180^\circ - 80^\circ \quad (\text{subtract } 80^\circ) \\ &= 100^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles on a straight line is equal to \(180^\circ\).$$\begin{aligned}60^\circ + x^\circ &= 180^\circ \\ x^\circ &= 180^\circ - 60^\circ \quad (\text{subtract } 60^\circ) \\ &= 120^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles on a straight line is equal to \(180^\circ\).$$\begin{aligned}x^\circ + 80^\circ + 60^\circ &= 180^\circ \\ x^\circ &= 180^\circ - 80^\circ - 60^\circ \quad (\text{subtract } 80^\circ \text{ and } 60^\circ) \\ &= 40^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles on a straight line is equal to \(180^\circ\).$$\begin{aligned}50^\circ + 75^\circ + x^\circ &= 180^\circ \\ x^\circ &= 180^\circ - 50^\circ - 75^\circ \quad (\text{subtract } 50^\circ \text{ and } 75^\circ) \\ &= 55^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles on a straight line is equal to \(180^\circ\).$$\begin{aligned}60^\circ + x^\circ + 55^\circ &= 180^\circ \\ x^\circ &= 180^\circ - 60^\circ - 55^\circ \quad (\text{subtract } 60^\circ \text{ and } 55^\circ) \\ &= 65^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles on a straight line is equal to \(180^\circ\). The three angles are equal (\(x^\circ\)).$$\begin{aligned}x^\circ + x^\circ + x^\circ &= 180^\circ \\ 3x^\circ &= 180^\circ \quad (\text{combine like terms}) \\ x^\circ &= 180^\circ \div 3 \quad (\text{divide by } 3) \\ &= 60^\circ\end{aligned}$$

Calculating an Unknown Angle in a Full Angle

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles in a point is equal to \(360^{\circ}\).$$\begin{aligned}200^\circ + x^\circ &= 360^\circ \\ x^\circ &= 360^\circ - 200^\circ \quad (\text{subtract } 200^\circ) \\ &= 160^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles around a point is equal to \(360^{\circ}\).$$\begin{aligned}x^\circ + 260^\circ &= 360^\circ \\ x^\circ &= 360^\circ - 260^\circ \quad (\text{subtract } 260^\circ) \\ &= 100^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles around a point is equal to \(360^{\circ}\).$$\begin{aligned}x^\circ + 80^\circ + 160^\circ &= 360^\circ \\ x^\circ &= 360^\circ - 80^\circ - 160^\circ \quad (\text{subtract } 80^\circ \text{ and } 160^\circ) \\ &= 120^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles around a point is equal to \(360^{\circ}\).$$\begin{aligned}80^\circ + x^\circ + 170^\circ &= 360^\circ \\ x^\circ &= 360^\circ - 80^\circ - 170^\circ \quad (\text{subtract } 80^\circ \text{ and } 170^\circ) \\ &= 110^\circ\end{aligned}$$

Exercise
Calculate the measure of the unknown angle.

\(x^{\circ}=\)\(^{\circ}\)

The sum of angles around a point is equal to \(360^{\circ}\). The three angles are equal (\(x^\circ\)).$$\begin{aligned}x^\circ + x^\circ + x^\circ &= 360^\circ \\ 3x^\circ &= 360^\circ \quad (\text{combine like terms}) \\ x^\circ &= 360^\circ \div 3 \quad (\text{divide by } 3) \\ &= 120^\circ\end{aligned}$$