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Here you will learn about angles, including parts of angles, measuring angles, types of angles, and special pairs of angles.

Students first learn about angles in 4th grade with their work in geometric measurements. They expand that knowledge as they progress through middle school.

**Angles** are formed where two rays or lines meet at a common point. The common point is called the vertex of the angle and the rays are called the arms of the angle.

The word angle comes from the Latin word “angulus” meaning corner.

Angles can be named by the points that exist on the arms and vertex.

For example,

This angle is named ‘ABC’ with the vertex letter in the middle.

\hspace{2.8cm} **Different Types of Angles**

Type of Angle | Description | Diagram |
---|---|---|

Acute angle | An angle whose measure is less than 90^{\circ} but greater than 0^{\circ} | |

Right angle | An angle that measure exactly 90^{\circ}. Right angles will have a square like symbol showing that they are equal to 90^{\circ} | |

Obtuse angle | An angle whose measure is greater than 90^{\circ} but less than 180^{\circ} | |

Straight angle | An angle whose measure is equal to 180^{\circ} | |

Reflex angle | An angle whose measure is greater than 180^{\circ} and less than 360^{\circ} |

**Step-by-step guide:** Types of angles

**Step-by-step guide:** Acute angles

**Step-by-step guide:** Obtuse angles

**Step-by-step guide:** Right angles

**Step-by-step guide:** Straight angle

Angles are measured in **degrees** and have the degree sign ^{\circ}. The tool used for measuring angles is called a **protractor**.

The degree measure on the top center of a protractor is 90^{\circ}. Notice that the numbers left and right of the center either go up by ten or go down by ten.

If the angle is acute, you will use the acute measurement, and if the angle is obtuse, you will use the obtuse measurement.

For example, let’s look at the measure of ∠ A.

The vertex, A, of the angle is placed on the bottom center of the protractor. One arm of the angle is lined up with the bottom of the protractor at 0^{\circ}. The other arm is used to measure the turn from one arm to the next.

Notice how the arm goes through the measure of 60^{\circ} and 120^{\circ}. Since the angle is obtuse, use the measurement that is greater than 90^{\circ} and less than 180^{\circ}. The angle measure 120^{\circ}.

**Step-by-step guide:** Measuring angles

**Angle rules** are facts that we can apply to calculate missing angles in a diagram.

The five key angle facts that are used widely within the topic are:

Angle Rule | Description | Diagram |
---|---|---|

Adjacent angles | Angles that share an arm and a vertex. Angle CBD and Angle ABD are adjacent. | |

Complementary angles | A pair of angles that have a sum of 90^{\circ}. These angles can be adjacent or not adjacent. If any two angles sum to 90^{\circ} they are complementary. | |

Supplementary angles | The sum of supplementary angles is 180^{\circ}. These angles can be adjacent or not adjacent. If any two angles sum to 180^{\circ} they are supplementary. | |

Vertical angles | Angles that are equal in measurement and formed by intersecting lines. |

**Step-by-step guide:** Adjacent angles

**Step-by-step guide:** Complementary angles

**Step-by-step guide:** Supplementary angles

**Step-by-step guide:** Vertical angle theorem

How does this apply to 4th grade math and 7th grade math?

**Grade 4 – Measurement and Data (4.MD.C.5.a)**An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.

**Grade 4 – Measurement and Data (4.MD.C.5.b**)

An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

**Grade 4 – Measurement and Data****(4.MD.C.6)**

Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

**Grade 4 – Measurement and Data (4.MD.C.7)**

Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, for example, by using an equation with a symbol for the unknown angle measure.

**Grade 4 – Geometry (4.G.A.1)**

Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

**Grade 7 – Geometry (7.G.B.5)**

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

In order to identify parts of an angle:

**Name the vertex.****Name the arms as rays.**

There are a lot of ways to use angles. For more specific step-by-step guides, check out the individual pages linked in the “What are angles?” section above or read through the examples below.

Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!

DOWNLOAD FREEUse this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!

DOWNLOAD FREEName the arms and vertex of the angle.

**Name the vertex.**

Point M is the vertex because it’s the point where the two rays meet.

2**Name the arms as rays.**

Ray ML is one arm and the other arm is ray MN.

What is the measure of the angle?

**Determine the type of angle.**

Angle A is an acute angle because it appears to be less than 90^{\circ}.

**Check to make sure the vertex is at the center of the protractor and one arm is lined up with the bottom of the protractor.**

The vertex which is point A is placed correctly on the protractor. The arm is lined up correctly on the bottom of the protractor.

**Find the degree measure.**

The other arm goes through the 50^{\circ} and 130^{\circ}. Since ∠A is acute, the correct measure of ∠A is 50^{\circ}.

Classify ∠ OMN as either acute, obtuse, right or straight.

**Recall the definitions of the types of angles needed.**

An acute angle is an angle that has a measure greater than 0^{\circ} and less than 90^{\circ}.

An obtuse angle is an angle that has a measure that is greater than 90^{\circ} and less than 180^{\circ}.

A right angle is equal to 90^{\circ} .

A straight angle is equal to 180^{\circ} .

**Explain how the angle fits the definition.**

∠ OMN is 107^{\circ}, so it is an obtuse angle because the measure is between 90^{\circ} and 180^{\circ}.

Classify the ∠ ABC and ∠ CBD as supplementary angles or complementary angles.

**Recall the definitions of the special pairs angles needed.**

Supplementary angles are a pair of angles whose sum is 180^{\circ}.

Complementary angles are a pair of angles whose sum is 90^{\circ}.

**Explain how the angle pair fits the definition.**

∠ ABC is 131^{\circ}

∠ CBD is 49^{\circ}

131+49=180

The sum of the angles is 180^{\circ} so they are supplementary angles.

The pair of angles are supplementary. Find the measure of Angle B.

**Recall the definition of the special pairs of angles.**

Supplementary angles are a pair of angles that when added together equal 180^{\circ}.

**Find the missing angle.**

∠ A is 27^{\circ}. To find ∠ B subtract 27 from 180.

180-27=153

∠ B is 153^{\circ} .

The pair of angles are complementary. Find the measure of angle QRT.

**Recall the definition of the special pairs of angles.**

Complementary angles are a pair of angles that when added together equal 90^{\circ} .

**Find the missing angle.**

∠ TRS is 62^{\circ}. To find ∠ QRT , subtract 62 from 90.

90-62=28

∠ QRT is 28^{\circ} .

- Give students hand held protractors so that they can learn how to use a protractor to measure angles.

- Have students draw different types of angles using colored pencils and then measure them so that they can practice how to measure angles while also being creative.

- Instead of giving students practice worksheets, have them sketch and measure the different angle pairs instead to help them formulate their own understanding.

**Measuring angles with a protractor when the diagrams are not drawn accurately**

When the question states that the diagram is not drawn accurately, we cannot simply measure the missing angles using a protractor. We need to use angle facts to calculate the missing angle.

**Confusing the definitions of the types of angles**

For example, thinking that an obtuse angle is one whose measure is between 0^{\circ} and 90^{\circ} instead of between 90^{\circ} and 180^{\circ}.

**Confusing the sum of complementary and supplementary angles**

For example, thinking that complementary angles sum to 180^{\circ} and that supplementary angles sum to 90^{\circ} when, in fact, complementary angles have a sum of 90^{\circ} and supplementary angles have a sum of 180^{\circ}.

1. What is the vertex of the angle?

O

OM

N

M

The vertex of an angle is where the two rays meet, or where the two arms meet.

In this case, point N is where the two rays meet.

2. Choose the correct classification of the ∠ A.

acute angle

obtuse angle

straight angle

reflex angle

The definition of an acute angle is an angle whose measure is between 0^{\circ} and 90^{\circ}.

∠ A is less than 90^{\circ} , so it is an acute angle.

3. Choose the correct classification of the angle.

acute angle

right angle

straight angle

reflex angle

The definition of a straight angle is an angle that measures 180^{\circ}.

On the protractor, you can see that the angle measures 180^{\circ} , so it is a straight angle.

4. Choose the correct classification of the angle.

acute angle

obtuse angle

straight angle

right angle

The definition of a right angle is an angle that measures 90^{\circ}.

On the protractor, you can see that the angle measures 90^{\circ} , so it is at a right angle.

5. ∠ A and ∠ B are complementary angles. Find the measure of ∠ B.

37^{\circ}

47^{\circ}

127^{\circ}

137^{\circ}

Complementary angles are a pair of angles whose sum is 90^{\circ}.

∠ A is 53^{\circ} , so to find ∠ B subtract 53 from 90.

90-53=37

∠ B is 37^{\circ} .

6. ∠ J and ∠ K are supplementary angles. Find ∠ J.

46^{\circ}

44^{\circ}

124^{\circ}

134^{\circ}

Supplementary angles are a pair of angles whose sum is 180^{\circ}.

∠ K is 46^{\circ} , so to find ∠ J subtract 46 from 180.

180-46=134

∠ J is 134^{\circ} .

The angle measurements you will be working with in elementary school and middle school are positive angle measurements. When you progress into high school you will work with negative angle measurements.

Understanding the types of angles will help you understand the angles of a triangle. Triangles are classified by their angles and their sides.

No, radians can be used as a unit of measurement for angles. However, you will not use radian measurements until high school.

Special angle pairs are formed from parallel lines and a transversal line intersecting the parallel lines. One of those special pair of angles formed is called alternate interior angles. There are other angles that are formed too such as alternate exterior angles and corresponding angles.

A straight angle always forms a straight line.

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