Angles

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.

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.

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.

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.

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.

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

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

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

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

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

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 \(50^\circ\).

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 \(30^\circ\).

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 \(100^\circ\).

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 \(90^\circ\).

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 \(120^\circ\).

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 \(115^\circ\).

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 \(45^\circ\).

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 \(50^\circ\).

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 \(30^\circ\).

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 \(100^\circ\).

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 \(90^\circ\).

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 \(120^\circ\).

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.

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.

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.

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.

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

• 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 \(180^\circ\).

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

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

• 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 \(180^\circ\).

• 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.

• 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.

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

• 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.

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.

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.

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.