Parallel angles

Here 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.

What are parallel angles?

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.

Properties of parallel lines

• Alternate interior angles are equal
  
  
  Sometimes called ‘Z angles’

• Corresponding angles are equal
  
  
  Sometimes called ‘F angles’

• Co-interior angles add up to \bf{180}^{\circ}
  
  
  Sometimes called ‘C angles’

Key angle facts

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

Parallel angles

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

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

Corresponding angles

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 (co-interior 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.

What are parallel angles?

Common Core State Standards

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 segment’s endpoints.

How to find a missing angle in parallel lines

In order to find a missing angle in parallel lines:

1. 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.
3. 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.

Parallel angles examples

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

Example 1: alternate angles

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

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

2State 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} .

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

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

Example 2: same side interior angles

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

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.

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}.

Example 3: corresponding angles

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

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.

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},

\begin{aligned}\theta &= 180-75 \\\\ \theta &= 105^{\circ}. \end{aligned}

Example 4: multi-step

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}.

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

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}.

Example 5: similar triangles

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}.

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

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.

Example 6: angles in parallel lines including algebra

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

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

\begin{aligned}& 4 x+30+20=180 \\\\ & 4 x=130 \\\\ & x=32.5^{\circ} \end{aligned}

Now that x=32.5^{\circ},

\begin{aligned}4x+30&=4 \times 32.5 + 30 \\\\ &=160^{\circ} \end{aligned}

Teaching tips for parallel angles

• 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.

Easy mistakes to make

• 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.

The next lessons are

• 2D shapes
• Triangles
• Quadrilateral
• Trigonometry

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