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Types of angles Supplementary angles Adjacent angles Perpendicular lines Polygons Intersecting linesHere you will learn how to recognize angles, use angle facts to find missing angles, and apply parallel angle facts to solve algebraic problems.

Students will first learn about parallel angles as a part of geometry in grade 8 and will expand on their learning throughout high school.

**Parallel angles** are angles that are created when a pair of parallel lines are intersected by a third line called a **transversal**.

You can use the information given in the diagram to find any angle around the intersecting transversal.

To do this, use three facts about angles in parallel lines: **Alternate Angles, Co-Interior Angles,** and **Corresponding Angles.**

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

DOWNLOAD FREE**Alternate interior angles are equal**

Sometimes called βZ anglesβ

**Corresponding angles are equal**

Sometimes called βF anglesβ

**Co-interior angles add up to \bf{180}^{\circ}**

Sometimes called βC anglesβ

To explore parallel angles, you will need to use some key angle facts.

It is known that vertically opposite angles are equal and can be shown around a point within parallel lines:

If the transversal line is extended so that it crosses more parallel lines, the angles are maintained throughout the diagram for any line parallel to the original line AB.

NOTE: For the same intersecting transversal, all the acute angles are the **same size**, and all the obtuse angles are the **same size**.

You can group these angles into three separate types called **alternate angles**,** same side interior angles** and **corresponding angles**.

**Alternate angles** are angles that occur on opposite sides of the transversal line and have the **same size.**

Each pair of **alternate angles** around the transversal are **equal** to each other. The two angles can either be a pair of alternate interior angles or alternate exterior angles.

Other examples of alternate angles:

You can often spot interior alternate angles by drawing a Z shape.

**Step by step: **Alternate interior angles theorem

The pairs of angles formed on the same side of the transversal that are either both obtuse or both acute are called corresponding angles and are equal in size.

Each pair of **corresponding angles** on the same side of the intersecting transversal is **equal** to each other.

Other examples of corresponding angles:

You can often spot interior corresponding angles by drawing an F shape.

**Step by step: **Corresponding angles

Same side interior angles, or consecutive interior angles occur in between two parallel lines when they are intersected by a transversal. The two angles that occur on the same side of the transversal always add up to 180^{\circ} .

Other examples of same side interior angles:

You can often spot same side interior angles by drawing a C shape.

**Step by step: **Same side interior angles

You may need to apply one or multiple different angle rules to work out missing angles in parallel lines.

It is important to note that most problems **do not have a unique method** to obtain a solution so it is crucial to be **very clear** with your working out.

How does this relate to 8 th grade math and high school math?

**Grade 8: Geometry (8.G.A.5)**Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

**High School: Geometry (HS.G.CO.C.9)**

Prove theorems about lines and angles.

Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segmentβs endpoints.

In order to find a missing angle in parallel lines:

**Highlight the angle(s) that you already know.****State the alternate angle, co-interior angle or corresponding angle fact to find a missing angle in the diagram.****Use basic angle facts to calculate the missing angle.**

**NOTE:** Steps 2 and 3 may be done in either order and may need to be repeated. Step 3 may not always be required.

For each stage of the calculation, you must clearly state any angle facts that are used.

Calculate the size of the missing angle \theta. Justify your answer.

**Highlight the angle(s) that you already know.**

2**State the alternate angle, co-interior angle or corresponding angle fact to find a missing angle in the diagram.**

Here you can label the alternate angle on the diagram as 50^{\circ} .

3**Use a basic angle fact to calculate the missing angle.**

Here as \theta is on a straight line with 50^{\circ},

\begin{aligned}\theta &= 180-50 \\\\ \theta &= 130^{\circ} \end{aligned}Calculate the size of the missing angle \theta. Justify your answer.

**Highlight the angle(s) that you already know.**

Here you can label the co-interior angle on the diagram as 60^{\circ} as 180-120=60^{\circ}.

**Use a basic angle fact to calculate the missing angle.**

You can see that as \theta is vertically opposite to 60^{\circ}, therefore \theta=60^{\circ}.

Calculate the size of the missing angle \theta. Justify your answer.

**Highlight the angle(s) that you already know.**

Here you can label the corresponding angle on the diagram as 75^{\circ}.

**Use a basic angle fact to calculate the missing angle.**

Here as \theta is on a straight line with 75^{\circ},

Calculate the missing angle \theta. Show all your work.

**Highlight the angle(s) that you already know.**

**Use a basic angle fact to calculate a missing angle.**

Opposite angles are equal so you can label the angle 110^{\circ}.

Co-interior angles add up to 180^{\circ}. Here 180-110=70^{\circ}.

\theta is corresponding to 70+35 so \theta=70+35=105^{\circ}.

Show that the two triangles are similar.

**Highlight the angle(s) that you already know.**

**Use a basic angle fact to calculate a missing angle.**

Here, you can see that the two angles highlighted in green are on a straight line and so their sum is 180^{\circ}. This gives us the missing angle of 70^{\circ}.

You can also see there are vertically opposite angles at the center of the diagram, both are 90^{\circ}.

The smaller triangle now has a missing angle of 20^{\circ} as angles in a triangle add to equal 180^{\circ}.

By stating the alternate angles to 70^{\circ} and 20^{\circ} you can see that \theta=20^{\circ} and the other angle in the triangle is 70^{\circ}. The two triangles contain the same angles and are therefore similar.

Given that the sum of angles on a straight line is equal to 180^{\circ}, calculate the value of x. Calculate the size of angle 4x+30.

**Highlight the angle(s) that you already know.**

State that 20^{\circ} is corresponding to the original angle.

**Use a basic angle fact to calculate the missing angle.**

As the sum of angles on a straight line is 180^{\circ}, you have

Now that x=32.5^{\circ},

- Students should be familiar with the basic concepts before attempting to work with parallel angles. Make sure students understand lines, rays, and the different types of angles.

- If using worksheets to provide practice to students, make sure to provide different levels of complexity to challenge students.

**Mixing up angle facts**

There are a lot of angle facts and it is easy to mistake alternate angles with corresponding angles. To prevent this from occurring, think about the alternate angles being on the alternate sides of the line.

**Using a protractor to measure an angle**

Most diagrams are not to scale so using a protractor will not result in a correct answer unless it is a coincidence.

1. Calculate the size of angle \theta.

\theta = 100^{\circ}

\theta =80^{\circ}

\theta =138^{\circ}

\theta =42^{\circ}

Using corresponding angles, the angle measures 42^{\circ}.

Then use angles on a straight line:

\theta=180-42= 138^{\circ}

2. Calculate the size of angle \theta.

\theta = 62^{\circ}

\theta =118^{\circ}

\theta =298^{\circ}

\theta =28^{\circ}

Using co-interior angles, calculate 180-62=118^{\circ}.

Label the corresponding angle 118^{\circ}.

Since opposite angles are equal,

\theta=118^{\circ}

3. Calculate the angle \theta.

\theta =21^{\circ}

\theta =159^{\circ}

\theta =69^{\circ}

\theta =79.5^{\circ}

Using opposite angles, the angle measures 21^{\circ}.

Next, label the alternate angle 21^{\circ}.

Then, use the fact that it is an isosceles triangle and so two angles are equal:

\theta=\cfrac{180-21}{2}=79.5^{\circ}.

4. Calculate theΒ size of angle \theta.

\theta =23^{\circ}

\theta =67^{\circ}

\theta =113^{\circ}

\theta =157^{\circ}

Using angles on a straight line, calculate 180-(90+67)=23^{\circ}.

Use alternate angles to find \theta=23^{\circ}.

5. Calculate the size of angle \theta.

\theta =151^{\circ}

\theta =88^{\circ}

\theta =121^{\circ}

\theta =59^{\circ}

Using angles on a straight line, calculate the angles 92^{\circ} and 59^{\circ}.

Then the other angle in the triangle is 180-(92+59)=29^{\circ}.

Using angles on a straight line, calculate 180-29=151^{\circ}.

Finally, using corresponding angles, \theta=151^{\circ}

6. By calculating the value of x, find the value of \theta.

\theta =150^{\circ}

\theta =90^{\circ}

\theta =30^{\circ}

\theta =85^{\circ}

30x-25 and 20x+5 are alternate angles, therefore, you can write 30x-25=20x+5

Then solve to find x\text{:}

\begin{aligned}30x-25&=20x+5 \\\\ 10x-25&=5 \\\\ 10x&=30 \\\\ x&=3 \end{aligned}

Given that x=3,

30 \times 3-25=65

Using opposite angles, notice that the angle inside the triangle is 65^{\circ}.

Using angles in a triangle, calculate the third angle in the triangle: 180-(65+30)=85^{\circ}.

Then using opposite angles, \theta=85^{\circ}.

Parallel angles are the types of angles that are formed when two parallel lines are cut by a transversal line. These types of angles include corresponding angles, alternate interior angles, alternate exterior angles and same side interior angles.

Parallel lines are two lines that will never intersect and are always an equal distance apart from each other. A transversal line is a line that crosses at least two other lines.

When a transversal line crosses parallel lines, several types of angles are created, including corresponding angles, alternate interior angles, and alternate exterior angles.

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