Same side interior angles

Here you will learn about same side interior angles. You’ll learn how to recognize same side interior angles, and apply this understanding to solve problems.

Students will first learn about same side interior angles as part of geometry in 8 th grade.

What are same side interior angles?

Same side interior angles, also known as consecutive interior angles or co-interior angles, are pairs of angles that are on the same side of a transversal line and inside the two lines that the transversal intersects.

The two angles that occur on the same side of the transversal are supplementary, so they always add up to 180^{\circ}.

The two interior angles are only equal when they are both 90^{\circ}.

You can find the angle measure of one of the same side interior angles by subtracting the other from 180^{\circ}.

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

What are same side interior 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 – Congruence (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 calculate with same side interior angles

In order to find a missing angle in parallel lines:

1. Highlight the angle(s) that you already know.
2. Use same side interior angles to find a missing angle.
3. Use basic angle facts if needed to calculate other missing angles.

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

Same side interior angles examples

Example 1: same side interior angles

Calculate the size of the missing angle x. Justify your answer.

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

The given angle is 113^{\circ}.

2Use same side interior angles to find a missing angle.

Here, you can label the co-interior angle on the diagram as 60^{\circ} as 113+67=180.

This gives you the measure of the missing angle:

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.

The given angle is 120^{\circ}.

Use same side interior angles to find a missing angle.

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

Use basic angle facts if needed to calculate other missing angles.

Since \theta and 60^{\circ} are vertically opposite angles, \theta=60^{\circ}.

Example 3: same side interior angles

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

Highlight the angle(s) that you already know.

The given angles are 105^{\circ} and 86^{\circ}.

Use same side interior angles to find a missing angle.

As same side interior angles have a sum of 180^{\circ}, \, 180-86=94 so the missing angle \theta=94^{\circ}.

Example 4: same side interior angles

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

Highlight the angle(s) that you already know.

Here, you can use either the 118^{\circ} angle or the 83^{\circ} angle. For this example, we will use the 118^{\circ} angle as this explores an alternative method that we haven’t looked at yet.

Use same side interior angles to find a missing angle.

As same side interior angles have a sum of 180^{\circ}, \, 180-118=62^{\circ} so the missing angle highlighted is 62^{\circ}.

Use basic angle facts if needed to calculate other missing angles.

Here you have a trapezoid with the sum of interior angles equal to 360^{\circ}.

Using this fact, you can calculate the value of \theta\text{:}

\begin{aligned}\theta&=360-(118+62+83) \\\\ \theta&=97^{\circ} \end{aligned}

Example 5: same side interior angles with algebra

Find the value of x. Justify your answer.

Highlight the angle(s) that you already know.

Two angles are labeled but you need to find the value of x.

Use same side interior angles to find a missing angle.

As 3x+45 and 3x-15 are co-interior angles, you can state that

\begin{aligned}& 3 x+45+3 x-15=180 \\\\ & 6 x+30=180 \\\\ & 6 x=150 \\\\ & x=25^{\circ} \end{aligned}

Example 6: same side interior angles with algebra

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

Highlight the angle(s) that you already know.

You need to find the value of x first.

Use same side interior angles to find a missing angle.

As 11x and 4x are co-interior angles, you can state that

\begin{aligned}11x+4x&=180 \\\\ 15x&=180 \\\\ x&=12^{\circ} \end{aligned}

This gives us the two same side interior angles of 11x=132^{\circ} and 4x=48^{\circ}.

Use basic angle facts if needed to calculate other missing angles.

The angles in the triangle are 48^{\circ} and 90^{\circ}, since the sum of angles on a straight line is 180^{\circ}.

180-(48+90)=42^{\circ} so the third angle in the triangle is 42^{\circ}.

Since \theta and 42^{\circ} are on a straight line,

\begin{aligned}& \theta=180-42 \\\\ & \theta=138^{\circ} \end{aligned}

Teaching tips for same side interior angles

• Begin by reviewing key angle relationships, such as complementary angles, supplementary angles, vertical angles, and adjacent angles. (Review can be given using a study guide or video lesson.) This helps students understand how same side interior angles fit into these broader categories, particularly their supplementary nature when lines are parallel.

• Start with a clear diagram showing two parallel lines cut by a transversal. You can use graph paper to make parallel lines easy to identify. Highlight the same side interior angles with different colors to help students easily identify them.

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 same side interior angles being inside the C shape.

• Using a protractor to measure an angle
  Using a protractor to measure an angle. Most diagrams are not to scale and so using a protractor will not result in a correct answer unless it is a coincidence.

• Confusing with polygon angles
  Mistaking angles in polygons, like those in a parallelogram, for same side interior angles. Same side interior angles specifically refer to angles formed by a transversal intersecting two lines, not just any interior angles within polygons.

Related parallel angles lessons

• Alternate interior angles theorem
• Corresponding angles

The next lessons are

• 2D shapes
• Triangles
• Quadrilateral

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